All practice questions

Question 401Question →

The state government is executing a 'Heritage Corridor Beautification' project that will pedestrianize a historically significant market street. As the Chief Project Officer, you are faced with conflicting demands from various interest groups. Match each stakeholder group's primary grievance to the most appropriate, proportionate administrative negotiation strategy designed to resolve their specific conflict.

Click a left item, then click its matching right item

Show answer & explanation

Answer

The correct matches are: Heritage Conservation Trust with real-time vibration monitoring; Street Vendors Association with phased relocation to temporary kiosks; Emergency Services Department with automated retractable bollards; and Market Retailers Union with a subsidized park-and-ride shuttle network.

Each administrative strategy directly addresses the specific root cause of the corresponding stakeholder's grievance, representing a 'win-win' negotiation approach. The solutions respect the core objective of the beautification project while accommodating the legitimate legal, economic, and safety concerns of the affected groups.

Step-by-Step Solution

1

Analyze the core grievance of the Heritage Conservation Trust.

Their concern is structural damage from heavy machinery.

Requires an administrative intervention that guarantees structural safety and builds trust through transparent oversight (real-time vibration monitoring).

2

Identify the most viable solution for the Local Street Vendors Association.

Match with phased relocation and future integration guarantees.

This strategy addresses their immediate need for daily livelihood while establishing a formal agreement for their long-term status.

3

Address the Emergency Services Department's critical operational requirement.

Match with automated retractable bollards featuring priority overrides.

This physical infrastructure solution maintains the pedestrian nature of the corridor without compromising rapid emergency response.

4

Resolve the Market Retailers Union's economic access concern.

Match with the subsidized 'park-and-ride' electric shuttle network.

This provides an alternative accessibility solution to offset the removal of adjacent street parking, sustaining customer footfall.

Key Concept

Proportionate Stakeholder Conflict Resolution

Question 402Question →

A group of students is standing in a single straight row facing South. Arjun is $26^{\text{th}}$ from the left end of the row, and Bhavya is $19^{\text{th}}$ from the right end of the row. When Arjun shifts $7$ places towards his left, he exactly occupies the middle position of the row. What is Bhavya's position from the left end of the row?

47^{\text{th}}

19^{\text{th}}

46^{\text{th}}

18^{\text{th}}

Show answer & explanation

Answer: $47^{\text{th}}$

Answer

Bhavya is exactly

47^{\text{th}}

from the left end of the row.

By properly recognizing that facing South reverses anatomical left and right relative to the row's ends, we determine Arjun's new position is

26 + 7 = 33

. A middle position of

33

yields a total of

65

students. Bhavya's position from the left is then calculated by subtracting her right-end rank from the total and adding

1

(

65 - 19 + 1 = 47

).

Step-by-Step Solution

1

Determine Arjun's movement direction relative to the row.

Arjun shifts towards the right end of the row.

Since the row is facing South, a person's personal left hand points towards the right end (geographic East) from an external observer's perspective.

2

Calculate Arjun's new position from the left end.

Arjun's new position is

26 + 7 = 33^{\text{rd}}

from the left end.

He was initially

26^{\text{th}}

and moved

7

places further away from the left end.

3

Determine the total number of students in the row (

N

).

The total number of students is

65

.

Since the

33^{\text{rd}}

position is exactly in the middle, the total is calculated as

N = (33 \times 2) - 1 = 65

.

4

Calculate Bhavya's position from the left end.

Bhavya is

47^{\text{th}}

from the left end.

Using the formula: Left Position = Total (

N

) - Right Position +

1

. Thus,

65 - 19 + 1 = 47

.

Key Concept

Directional awareness in linear arrangements and position reversal formulas.

Estimated Time:1m 15s

Question 403Question →

The state agriculture department recently issued the following directive:

"To halt the rapid depletion of groundwater reserves in the western districts, the cultivation of sugarcane is prohibited in these areas with immediate effect. A parallel financial incentive scheme will be launched for farmers transitioning to millet farming."

Based on the statement above, which of the following assumptions are implicit in the department's directive? (Select all valid assumptions)

Select all that apply

Millet farming consumes substantially less groundwater compared to sugarcane cultivation.

Providing financial incentives is an effective strategy to encourage farmers to adopt different agricultural practices.

The state's overall production of sugarcane will experience a severe decline in the upcoming harvest season.

Sugarcane cultivation is the sole cause of water scarcity across all geographical regions of the state.

Show answer & explanation

Answer: Millet farming consumes substantially less groundwater compared to sugarcane cultivation.; Providing financial incentives is an effective strategy to encourage farmers to adopt different agricultural practices.

Answer

The correct assumptions are that millet consumes less water than sugarcane, and that financial incentives will effectively encourage farmers to change their crops.

The correct assumptions identify the necessary conditions for the policy to work. For the substitution to halt water depletion, the new crop (millet) must logically use less water than the banned crop (sugarcane). Furthermore, offering financial support inherently assumes that such incentives are a viable way to drive behavioral change among the farmers.

Step-by-Step Solution

1

Analyze the primary goal and actions presented in the statement.

The goal is to halt groundwater depletion in western districts. The actions are banning sugarcane and offering financial incentives to grow millet.

Identifying the core components of the directive helps isolate what must be true for the directive to make logical sense.

2

Evaluate the premise behind the crop substitution strategy.

The department must assume that growing millet will use significantly less water than growing sugarcane.

If millet consumed the same or more water than sugarcane, the substitution would fail to achieve the primary goal of saving groundwater.

3

Evaluate the premise behind the financial strategy.

The department must assume that offering money will persuade farmers to comply and switch crops.

Policies utilizing financial incentives inherently assume that those incentives are an effective catalyst for behavioral change.

4

Differentiate between implicit assumptions and logical inferences.

A drop in future sugarcane production is a consequence (inference), not an underlying premise (assumption). Broad generalizations about the entire state's water scarcity are external exaggerations.

Assumptions must be foundational pillars required before the statement is made, not outcomes that happen after the statement is executed.

Key Concept

Statement and Assumptions

Question 404Question →

A numerical access code $X$ is generated based on the evaluation of the following mathematical expression:

X = 36 + 144 \div 6 \times 4

To successfully bypass the security protocol, a user must input the exact total number of positive factors (divisors) of the value $X$ . What number must the user input?

3

8

12

16

Show answer & explanation

Answer: 12

Answer

The user must input 12, as the evaluated number 132 possesses exactly 12 positive factors.

To find the correct access code, the expression must first be properly evaluated using standard order of operations (BODMAS). By performing division and then multiplication left-to-right, we get

144 \div 6 = 24

, and

24 \times 4 = 96

. Adding 36 yields

X = 132

. The prime factorization of 132 is

2^2 \times 3^1 \times 11^1

. Using the standard formula for finding total divisors, we add 1 to each exponent and multiply them together:

(2+1)(1+1)(1+1) = 3 \times 2 \times 2 = 12

.

Step-by-Step Solution

1

Evaluate the division and multiplication parts of the expression from left to right according to the BODMAS rule.

144 \div 6 = 24

, followed by

24 \times 4 = 96

.

Division and multiplication hold equal precedence and must be executed left-to-right before any addition.

2

Complete the addition to find the final integer value of

X

.

X = 36 + 96 = 132

.

Addition is performed after higher-precedence operations are resolved.

3

Determine the prime factorization of the computed value 132.

132 = 4 \times 33 = 2^2 \times 3^1 \times 11^1

.

Finding the prime bases and their respective powers is required to apply the divisor count formula.

4

Calculate the total number of positive factors by adding 1 to each exponent and multiplying the results.

(2 + 1) \times (1 + 1) \times (1 + 1) = 3 \times 2 \times 2 = 12

.

The formula

(a+1)(b+1)...

systematically counts every possible product combination of the prime factors.

Key Concept

Factors, Multiples, and Prime Factorization

Estimated Time:1m 30s

Question 405Question →

An integer $P$ is given by the mathematical expression $P = 4 \cdot 36^{2033} - 6 \cdot 58^{2033} - 2 \cdot 89^{2033}$ . Determine the positive remainder when $P$ is divided by $17$ .

Show answer & explanation

Answer: 9

Answer

9

The correct answer is derived by first reducing the bases modulo 17, giving 2, 7, and 4 respectively. Then, utilizing Fermat's Little Theorem (

a^{16} \equiv 1 \pmod{17}

), the exponent 2033 is reduced to

2033 \pmod{16} = 1

. Substituting these simplified values back into the expression yields

4(2) - 6(7) - 2(4) = -42

. Because a remainder must be positive, adding the next highest multiple of 17 (which is

17 \times 3 = 51

) to -42 gives the final valid remainder of 9.

Step-by-Step Solution

1

Find the remainder of each base when divided by 17.

36 \equiv 2 \pmod{17}

,

58 \equiv 7 \pmod{17}

, and

89 \equiv 4 \pmod{17}

.

Reducing bases before dealing with large exponents simplifies the modular arithmetic calculation.

2

Apply Fermat's Little Theorem to simplify the exponent.

Since 17 is a prime number,

a^{16} \equiv 1 \pmod{17}

. The exponent is

2033 = 16 \times 127 + 1

, meaning

a^{2033} \equiv a^1 \pmod{17}

.

Fermat's Little Theorem allows us to reduce massive exponents by dividing them by

(p-1)

and keeping only the remainder.

3

Substitute the reduced bases and exponents into the original expression.

P \equiv 4(2^1) - 6(7^1) - 2(4^1) \pmod{17}

, which evaluates to

8 - 42 - 8 = -42

.

This step calculates the combined overall remainder before adjusting for the strict definition of a positive modulo.

4

Convert the negative result to a valid positive remainder.

-42 \equiv -42 + 51 = 9 \pmod{17}

.

A remainder must be a non-negative integer strictly less than the divisor (17). Adding the next highest multiple of 17 (which is 51) provides the correct positive remainder.

Key Concept

Divisibility Rules and Remainder Theorem

Question 406Question →

In a centralized database architecture, data records are distributed across $17$ server nodes numbered from $0$ to $16$ . A record with a numerical key $K$ is assigned to a node using the hash function $H(K) = K \pmod{17}$ , which always yields a positive remainder. If a specific batch of records is assigned a master key given by the expression $K = 2^{200} - 75$ , which server node will process this batch?

Show answer & explanation

Answer: 11

Answer

The batch of records will be processed by server node 11.

By applying modular arithmetic rules, the expression

(2^{200} - 75) \pmod{17}

simplifies individually to

(1 - 7) = -6

. Because node numbers in this system must be positive, adding the divisor 17 to -6 provides the correct positive remainder, which is 11.

Step-by-Step Solution

1

Identify the mathematical goal of the problem.

Calculate

(2^{200} - 75) \pmod{17}

and ensure the final result is a positive integer between 0 and 16.

The server nodes are assigned strictly based on the positive remainder when the key is divided by 17.

2

Find the remainder of the exponential term

2^{200}

divided by 17.

2^4 = 16 \equiv -1 \pmod{17}

. Raising this to the 50th power gives

(2^4)^{50} \equiv (-1)^{50} = 1 \pmod{17}

.

Finding a power of the base that is 1 or -1 modulo the divisor simplifies extremely large exponent calculations.

3

Find the remainder of the constant term 75 divided by 17.

75 = 17 \times 4 + 7

, so

75 \equiv 7 \pmod{17}

.

Each term in a modular arithmetic expression can and should be reduced individually.

4

Combine the reduced terms to find the overall remainder.

1 - 7 = -6 \pmod{17}

.

Substitute the individual reduced remainders back into the original expression

K = 2^{200} - 75

.

5

Convert the negative remainder into a valid positive node number.

-6 + 17 = 11

.

A remainder of -6 indicates the value is 6 units short of a full multiple of 17. The equivalent positive remainder is found by adding the divisor.

Key Concept

Modular arithmetic with large exponents and handling negative remainders appropriately.

Estimated Time:1m 30s

Question 407Question →

Align each of the following governance dilemmas with the primary ethical or administrative priority that should dictate the officer's response.

Click a left item, then click its matching right item

Show answer & explanation

Answer

The correct alignment pairs the riot intelligence with preemptive public order, the vaccine protocol issue with substantive public health, the storm-drain settlement with disaster mitigation, and the political pressure with prosecutorial independence.

Each dilemma requires an administrator to weigh competing interests. The correct matches identify the overriding public interest in each scenario: safety over ceremony, health over paperwork, city-wide disaster prevention over unauthorized settlements, and legal integrity over politics.

Step-by-Step Solution

1

Evaluate the riot intelligence dilemma.

The primary threat is to life and general peace.

This necessitates 'Preemptive Public Order and Security'.

2

Evaluate the vaccine protocol dilemma.

The immediate need to save lives outweighs administrative paperwork.

This aligns with 'Substantive Public Health over Procedural Rigidity'.

3

Evaluate the storm drain settlement dilemma.

The encroachment threatens the city with impending floods.

The correct priority is 'Preventive Disaster Mitigation'.

4

Evaluate the political pressure dilemma.

The integrity of the justice system is at stake.

The appropriate principle is 'Prosecutorial Independence and Neutrality'.

Key Concept

Prioritizing administrative actions based on situational ethics and the greatest public good.

Question 408Question →

You are the Municipal Commissioner of a metropolitan city introducing a mandated mechanized solid waste management system. An unorganized but large collective of informal sanitation workers has blockaded the municipal headquarters, fearing the new mechanized sorting facilities will destroy their traditional livelihoods. Simultaneously, prominent Resident Welfare Associations (RWAs) are threatening to immediately withhold their property tax payments because garbage has been accumulating on the streets for three days. To resolve this stakeholder conflict effectively, which of the following is the most appropriate initial administrative strategy?

Invite the sanitation worker representatives for negotiations to discuss integrating them into the formalized mechanized system, while deploying emergency municipal reserve staff to clear waste from critical residential zones.

Request immediate police intervention to disperse the blockade at the headquarters and arrest the protest leaders to ensure the mechanized system is implemented without further delay.

Immediately suspend the implementation of the mechanized waste system and revert entirely to manual collection to appease the workers and prevent the looming RWA tax boycott.

Issue a final, non-negotiable public ultimatum requiring the workers to clear the streets within four hours or face permanent blacklisting from all future municipal employment.

Show answer & explanation

Answer: Invite the sanitation worker representatives for negotiations to discuss integrating them into the formalized mechanized system, while deploying emergency municipal reserve staff to clear waste from critical residential zones.

Answer

The most appropriate strategy is to invite the worker representatives for negotiations to discuss formal integration while simultaneously using emergency reserve staff to clear critical waste.

The correct answer demonstrates the optimal administrative balance. By deploying emergency staff, the administrator fulfills the immediate statutory duty to maintain public hygiene and pacifies the residents. Simultaneously, by opening negotiations focused on integrating the informal workers into the new formalized system, the administrator directly addresses the root cause of the conflict (livelihood insecurity) without abandoning the modernization mandate.

Step-by-Step Solution

1

Analyze the competing core interests of the primary stakeholders.

Identified that workers need livelihood security, RWAs require immediate public hygiene, and the municipality must implement the mandated modernization policy.

Effective conflict resolution requires understanding the fundamental needs driving each group's behavior.

2

Address the immediate crisis component without derailing long-term objectives.

Deploying emergency municipal reserves handles the acute public health risk and pacifies the RWAs temporarily.

Mitigating the most urgent pressure (accumulating garbage and tax boycotts) creates the necessary time and administrative breathing room for complex negotiations.

3

Initiate a collaborative negotiation framework focused on mutual gains.

Opening dialogue to formalize and integrate the informal workers into the new mechanized system transforms resistance into cooperation.

Win-win negotiation strategies that accommodate legitimate livelihood concerns ensure smoother policy implementation than coercion or capitulation.

Key Concept

Proportional Conflict Resolution and Stakeholder Integration

Question 409Question →

Five startup companies—Solaris, Terra, Umbra, Vesta, and Wyvern—are being ranked by their market valuation from highest to lowest. Based on the following information, arrange the companies in the correct order from highest valuation (top) to lowest valuation (bottom):

1. Umbra is valued higher than both Vesta and Wyvern.
2. Terra is valued lower than Solaris but higher than Umbra.
3. Wyvern is not the lowest valued company.

Drag items to arrange them in the correct order

Show answer & explanation

Answer

The correct order from highest to lowest valuation is: Solaris, Terra, Umbra, Wyvern, and Vesta.

Based on the provided clues, Solaris has the highest valuation, followed directly by Terra, and then Umbra. Since Umbra is valued higher than both Vesta and Wyvern, those two must occupy the fourth and fifth spots. Because Wyvern cannot be the lowest valued company, it must take the fourth position, leaving Vesta as the fifth and lowest valued company.

Step-by-Step Solution

1

Analyze the relationship between Terra, Solaris, and Umbra.

Solaris > Terra > Umbra

The second clue explicitly states that Terra is valued lower than Solaris but higher than Umbra.

2

Analyze the relationship between Umbra, Vesta, and Wyvern.

Umbra > Vesta and Umbra > Wyvern

The first clue states that Umbra is valued higher than both Vesta and Wyvern.

3

Combine the findings from Step 1 and Step 2 to form a partial sequence.

Solaris > Terra > Umbra > (Vesta and Wyvern)

Applying the transitive property, Solaris and Terra occupy the first and second positions, Umbra is third, leaving Vesta and Wyvern for the bottom two spots.

4

Determine the exact final positions of Vesta and Wyvern.

Wyvern > Vesta

The third clue states that Wyvern is not the lowest valued company. Therefore, between the remaining two spots, Wyvern must be fourth and Vesta must be fifth.

Key Concept

Relative position ordering using transitive logic and negative constraints.

Question 410Question →

Three automated data backup protocols in a cloud server execute network synchronization pulses at precise intervals. Protocol Alpha pulses every $\frac{18}{5}$ seconds, Protocol Beta every $\frac{24}{25}$ seconds, and Protocol Gamma every $\frac{36}{35}$ seconds. They all initiate their first pulse simultaneously at 08:00:00 AM. Assuming the initial pulse at 08:00:00 AM is excluded from the count, which of the following represents the total number of times all three protocols will pulse together in exactly $12$ minutes?

21000

10

50

51

Show answer & explanation

Answer: $50$

Answer

The data protocols will pulse together

50

times in the

12

-minute period.

To find when all three protocols pulse together, we must determine the Least Common Multiple (LCM) of their cycle times. The LCM of fractions is calculated by dividing the LCM of their numerators (

18, 24, 36

) by the HCF of their denominators (

5, 25, 35

). The LCM of

18, 24,

and

36

is

72

. The HCF of

5, 25,

and

35

is

5

. This gives a synchronized interval of

72/5 = 14.4

seconds. In

12

minutes (

720

seconds), the number of synchronized intervals is

720 \div 14.4 = 50

. Since the initial pulse is excluded as per the instructions, the correct answer remains

50

.

Step-by-Step Solution

1

Identify the mathematical concept required to find when all three protocols pulse simultaneously.

The time of simultaneous pulsing is the Least Common Multiple (LCM) of their individual cycle times:

\frac{18}{5}

,

\frac{24}{25}

, and

\frac{36}{35}

.

Simultaneous occurrence of periodic events always aligns at multiples common to all individual intervals.

2

Apply the formula for finding the LCM of fractions.

\text{LCM of fractions} = \frac{\text{LCM of numerators}}{\text{HCF of denominators}}

.

This is the standard algebraic rule to find the least common multiple for fractional values.

3

Calculate the LCM of the numerators (

18, 24, 36

) and the HCF of the denominators (

5, 25, 35

).

\text{LCM}(18, 24, 36) = 72

and

\text{HCF}(5, 25, 35) = 5

. Thus, the synchronized interval is

\frac{72}{5}

seconds, or

14.4

seconds.

Prime factorization gives

18 = 2 \times 3^2

,

24 = 2^3 \times 3

,

36 = 2^2 \times 3^2

(LCM =

2^3 \times 3^2 = 72

). For denominators,

5

is the largest common divisor.

4

Convert the given total time window into seconds.

12 \text{ minutes} = 12 \times 60 = 720 \text{ seconds}

.

Units must match the cycle interval (seconds) before division.

5

Calculate the total number of simultaneous pulses.

720 \div 14.4 = 50

. Since the initial pulse is excluded, the final count remains

50

.

Dividing the total time window by the synchronized interval yields the number of subsequent events.

Key Concept

Least Common Multiple (LCM) of fractions and its application to periodic simultaneous events.

Question 411Question →

You are a Municipal Commissioner. A team of municipal engineers reports that a newly built commercial complex has severe structural safety code violations, posing an imminent risk of collapse. They recommend immediate sealing of the building. However, the Mayor informally advises you to impose a token fine and allow the building to open in order to avoid political friction, as the construction company is highly influential.

Rank the following courses of action from the MOST ethically sound and administratively appropriate to the LEAST appropriate.

Drag items to arrange them in the correct order

Show answer & explanation

Answer

The most appropriate action is to immediately seal the building, followed by suspending the usage certificate with a show-cause notice. Fining the company while allowing usage is unethical, and completely ignoring the report is the worst possible action.

The most ethical course of action must uncompromisingly prioritize public safety when there is an imminent risk to life, making immediate sealing the best action. The next best action is temporarily suspending usage while following due process. Imposing a fine but allowing usage is highly unethical as it puts lives at risk for political convenience. Completely ignoring the report is the worst action, representing a total abdication of administrative responsibility.

Step-by-Step Solution

1

Analyze the core ethical dilemma and competing values.

The scenario presents a direct conflict between paramount public safety (engineers' report of imminent collapse) and undue political pressure (Mayor's advice to overlook the issue).

Identifying the competing interests is the essential first step in administrative ethical decision-making.

2

Evaluate each action based on the principle of public interest and statutory duty.

Immediate sealing addresses the imminent threat directly. Suspending usage and issuing a notice is procedurally sound but slightly less decisive. Fining while allowing usage directly endangers the public. Ignoring the report is a total failure of duty.

Administrative ethics dictates that protecting human life and public safety is the non-negotiable, paramount duty of a civil servant.

3

Order the actions from most to least aligned with administrative ethics.

The logical sequence is: Sealing (protects life, enforces law) -> Notice (protects life temporarily, follows procedure) -> Fine (endangers life, political compromise) -> Ignore (endangers life, corruption).

This establishes the final ranking based on the degree to which each action upholds or compromises ethical administrative principles.

Key Concept

Primacy of Public Safety and Neutrality in Public Administration

Question 412Question →

Read the following policy announcement carefully.

Statement:
"The municipal corporation has introduced an additional 5% property tax rebate for housing societies that successfully process 100% of their organic waste within their own premises."

Which of the following assumptions are logically implicit in the given statement? (Select all that apply)

Select all that apply

Financial incentives can effectively motivate housing societies to adopt and maintain in-house waste management systems.

Housing societies possess, or can acquire, the necessary spatial and technical capability to process all their organic waste internally.

The total volume of waste collected by municipal garbage trucks will decrease significantly over the next year.

Organic waste decomposes naturally and is generally less harmful to the environment than non-biodegradable electronic waste.

Show answer & explanation

Answer: Financial incentives can effectively motivate housing societies to adopt and maintain in-house waste management systems.; Housing societies possess, or can acquire, the necessary spatial and technical capability to process all their organic waste internally.

Answer

The implicit assumptions are that financial incentives can motivate housing societies, and that these societies have the practical capability to process their organic waste internally.

The correct answers identify the foundational premises required for the policy to be logical and executable. The policy offers a reward to encourage a specific action. For this policy to be coherent, the municipal corporation must assume both that the financial reward is an effective motivator and that the housing societies are practically capable of performing the required action.

Step-by-Step Solution

1

Identify the core subject and intent of the statement.

The municipality is offering a tax rebate (a financial reward) to housing societies for taking on the responsibility of processing their own organic waste.

Understanding the core mechanism of the statement is necessary to deduce the author's underlying thought process.

2

Apply the negation test to the first valid premise.

If financial incentives do NOT motivate housing societies, the policy is useless. Therefore, the assumption that they do motivate is implicit.

An assumption is a necessary foundational premise; without it, the argument or policy collapses.

3

Apply the negation test to the second valid premise.

If housing societies completely lack the space or capability to process waste, offering a rebate for it is illogical. Therefore, the assumption of capability is implicit.

Policies inherently assume the feasibility of the actions they seek to encourage.

4

Evaluate the remaining options against the definition of an assumption.

One remaining option is an inference (a future outcome of the policy), and the other is an irrelevant general fact.

It is critical to separate unstated foundational premises from logical conclusions and external truths.

Key Concept

Statement and Assumptions

Question 413Question →

Consider the positive integer $N = 3600$ . What is the total number of positive divisors of $N$ that are multiples of either $4$ or $9$ ?

12

42

33

9

Show answer & explanation

Answer: 33

Answer

The total number of positive divisors of 3600 that are multiples of either 4 or 9 is 33.

The correct answer is derived by finding the prime factorization of 3600 (

2^4 \times 3^2 \times 5^2

), determining the separate counts of divisors that are multiples of 4 (27 factors) and multiples of 9 (15 factors), and then using the inclusion-exclusion principle to subtract the intersection (divisors that are multiples of 36, which is 9 factors). This yields

27 + 15 - 9 = 33

.

Step-by-Step Solution

1

Prime factorize 3600 to identify the structure of its divisors.

3600 = 36 \times 100 = 2^4 \times 3^2 \times 5^2

.

A divisor's properties depend entirely on the combinations of its prime factors.

2

Calculate the total number of divisors that are multiples of 4.

Multiples of 4 must contain at least

2^2

. The valid powers for 2 are

\{2^2, 2^3, 2^4\}

(3 choices). For 3,

\{3^0, 3^1, 3^2\}

(3 choices). For 5,

\{5^0, 5^1, 5^2\}

(3 choices). Total =

3 \times 3 \times 3 = 27

divisors.

To be a multiple of 4, the divisor's prime factorization must contain a power of 2 that is at least 2.

3

Calculate the total number of divisors that are multiples of 9.

Multiples of 9 must contain at least

3^2

. The valid powers for 2 are 5 choices (from 0 to 4). For 3,

\{3^2\}

(1 choice). For 5, 3 choices. Total =

5 \times 1 \times 3 = 15

divisors.

To be a multiple of 9, the divisor must contain a power of 3 that is at least 2.

4

Calculate the number of divisors that are multiples of both 4 and 9 (i.e., multiples of 36).

These divisors must contain at least

2^2

and

3^2

. Choices: 3 (for 2)

\times

1 (for 3)

\times

3 (for 5) =

9

divisors.

Since 4 and 9 are co-prime, multiples of both must be multiples of their Least Common Multiple (LCM), which is 36.

5

Apply the Principle of Inclusion-Exclusion to find divisors that are multiples of either 4 or 9.

27 + 15 - 9 = 33

divisors.

Adding the multiples of 4 and 9 counts their intersection (multiples of 36) twice, so it must be subtracted once to find the correct union.

Key Concept

Applying set theory (inclusion-exclusion principle) to the combinations of prime factors of a number to find constrained subsets of divisors.

Question 414Question →

Calculate the remainder when the integer value of $11 \times 17^{73}$ is divided by $19$ .

Show answer & explanation

Answer: 16

Answer

16

The correct remainder is found by applying Fermat's Little Theorem to reduce the large exponent, yielding an intermediate calculation of -22. Converting this negative value to a proper positive remainder modulo 19 gives exactly 16.

Step-by-Step Solution

1

Simplify the base of the exponent modulo 19.

17 \equiv -2 \pmod{19}

Working with a smaller absolute value simplifies subsequent exponentiation steps.

2

Apply Fermat's Little Theorem to identify the cyclicity.

a^{18} \equiv 1 \pmod{19}

for any integer

a

not divisible by 19.

Since 19 is a prime number, the remainders of powers will repeat in cycles of 18.

3

Reduce the large exponent

73

using the identified cyclicity.

73 = 18 \times 4 + 1

, which means

(-2)^{73} \equiv (-2)^1 \pmod{19}

Because the powers cycle every 18, only the remainder of the exponent divided by 18 affects the final result.

4

Multiply the reduced exponential term by the leading coefficient.

11 \times (-2) = -22

The original mathematical expression contains a coefficient of 11.

5

Convert the negative intermediate result to a valid positive remainder.

-22 \equiv -22 + 2 \times 19 = -22 + 38 = 16 \pmod{19}

By standard definition, a remainder must be a positive integer strictly less than the divisor.

Key Concept

Modular Arithmetic and Fermat's Little Theorem

Question 415Question →

You are the Sub-Divisional Magistrate (SDM) overseeing land acquisition for a critical state irrigation canal. The deadline to hand over the acquired land to the engineering department is tomorrow. The state government is strictly monitoring the timeline. However, a group of marginalized farmers has not received their mandatory compensation due to a technical glitch in the centralized digital treasury system. They are peacefully protesting at the site, refusing to vacate until the compensation is credited. The engineering contractor demands that you use police force to clear the site immediately to avoid project delays. As an administrator, which of the following is the most appropriate course of action?

Temporarily pause the physical eviction, urgently coordinate with the treasury department to resolve the glitch, and communicate a concrete resolution timeline to the farmers.

Order the immediate deployment of police force to disperse the protesters and secure the site, ensuring the government's project deadline is strictly met.

Direct the engineering contractor to distribute equivalent cash payments to the farmers from their own funds to bypass the treasury delay.

Halt all land acquisition operations across the entire district indefinitely until the treasury software undergoes a comprehensive audit and upgrade.

Show answer & explanation

Answer: Temporarily pause the physical eviction, urgently coordinate with the treasury department to resolve the glitch, and communicate a concrete resolution timeline to the farmers.

Answer

Temporarily pause the physical eviction, urgently coordinate with the treasury department to resolve the glitch, and communicate a concrete resolution timeline to the farmers.

The correct answer balances empathy, administrative duty, and public order. It respects the statutory right of the farmers to receive their rightful compensation before displacement while actively utilizing official channels to resolve the systemic issue and resume the project as quickly as possible.

Step-by-Step Solution

1

Analyze the competing priorities in the scenario.

The conflict is between meeting a strict project deadline and respecting the legal rights of citizens who are victims of an administrative error.

Administrative decision-making requires weighing public interest against individual rights and procedural fairness.

2

Evaluate the demands of the engineering contractor.

The demand to use police force is rejected as disproportionate, given that the farmers are protesting peacefully for a legitimate grievance caused by the administration.

Force should be a last resort, never used to cover up administrative shortcomings.

3

Determine the most balanced and legally sound resolution.

The best course is to temporarily hold the eviction, fix the root cause (the treasury glitch) through proper channels, and maintain transparent communication with the affected farmers.

This upholds constitutional values, maintains public trust, and follows established statutory procedures without overreacting.

Key Concept

Administrative proportionality, due process, and stakeholder communication during procedural failures

Question 416Question →

A logistics manager is arranging shipping containers into equal stacks. When the containers are stacked in groups of $18$ , $24$ , or $32$ , there are always exactly $11$ containers left over. However, when they are stacked in groups of $35$ , there are no containers left over. What is the minimum total number of containers the manager could have?

Show answer & explanation

Answer: 875

Answer

875

The correct answer is derived by recognizing that the total number of containers must take the form of

LCM(18, 24, 32) \times k + 11

. The LCM is

288

, so the number is

288k + 11

. We then find the smallest integer

k

such that

(288k + 11)

is perfectly divisible by

35

. By evaluating the expression for sequential values of

k

(or using modular arithmetic,

8k \equiv 24 \pmod{35}

), we find that

k = 3

is the smallest valid multiplier. Substituting this back yields

288(3) + 11 = 875

.

Step-by-Step Solution

1

Calculate the Least Common Multiple (LCM) of the initial stack group sizes (

18

,

24

, and

32

).

The prime factorizations are

18 = 2 \times 3^2

,

24 = 2^3 \times 3

, and

32 = 2^5

. The LCM is

2^5 \times 3^2 = 32 \times 9 = 288

.

The base cycle for the stacks without remainders requires finding the smallest number perfectly divisible by all three group sizes.

2

Express the total number of containers algebraically.

Let the total number of containers be

N

. We can write

N = 288k + 11

, where

k

is a positive integer.

Since there is always a remainder of

11

containers when divided by these numbers, the total must be

11

more than a multiple of their LCM.

3

Apply the secondary divisibility condition.

The problem states that

N

is exactly divisible by

35

. Therefore,

288k + 11 \equiv 0 \pmod{35}

.

When the containers are grouped by

35

, there is no remainder.

4

Solve the congruence to find the smallest integer

k

.

First, simplify

288 \pmod{35}

: since

35 \times 8 = 280

, we have

288 \equiv 8 \pmod{35}

. The equation becomes

8k + 11 \equiv 0 \pmod{35}

, or

8k \equiv -11 \equiv 24 \pmod{35}

. Dividing both sides by

8

gives

k = 3

.

Finding the smallest valid positive integer

k

will yield the minimum total number of containers.

5

Calculate the final number of containers using

k = 3

.

N = 288(3) + 11 = 864 + 11 = 875

.

Substitute the multiplier back into the original algebraic expression.

Key Concept

Solving for an unknown quantity using the Least Common Multiple (LCM) combined with constant remainder logic and a secondary divisibility condition.

Alternative Method

Instead of using modular arithmetic to solve

288k + 11 \equiv 0 \pmod{35}

, you can manually test integer values for

k

in the formula

N = 288k + 11

. For

k=1

,

N = 288(1) + 11 = 299

(

299 \div 35

leaves remainder

19

). For

k=2

,

N = 288(2) + 11 = 587

(

587 \div 35

leaves remainder

27

). For

k=3

,

N = 288(3) + 11 = 875

(

875 \div 35 = 25

with no remainder). Therefore,

875

is the smallest valid number.

Estimated Time:2m 30s

Question 417Question →

A solar energy farm has a continuous linear array of $105$ solar panels. Panel Sigma is currently positioned $43^{\text{rd}}$ from the left end of the array. Due to a localized upgrade project, the first $18$ panels from the left end and the last $24$ panels from the right end are temporarily removed from the array. In the newly formed, shorter array of remaining panels, what is the position of Panel Sigma from the right end?

Show answer & explanation

Answer: 39

Answer

39

Panel Sigma's original position from the left is

43^{\text{rd}}

. When

18

panels are removed from the front, its new position from the left becomes

43 - 18 = 25^{\text{th}}

. The initial total was

105

panels. After removing

18

from the left and

24

from the right, the new total is

105 - 18 - 24 = 63

panels. Using the standard ranking formula (

Total = Left + Right - 1

), the position from the right is calculated as

63 - 25 + 1 = 39^{\text{th}}

.

Step-by-Step Solution

1

Calculate the new total number of panels in the array.

105 - 18 - 24 = 63

panels

Removing panels from both ends reduces the overall length of the array.

2

Determine Panel Sigma's new position from the left end.

43 - 18 = 25^{\text{th}}

from the left

Removing 18 panels from the left shifts Panel Sigma 18 positions closer to the left end.

3

Calculate Panel Sigma's new position from the right end.

Position from right =

63 - 25 + 1 = 39^{\text{th}}

Using the standard ranking formula (Total = Left Position + Right Position - 1), we find the right-side position.

Key Concept

Positional shift and total rank calculation in a truncated linear arrangement

Alternative Method

Instead of calculating the new total, you can determine how many panels are strictly to the right of Panel Sigma. Initially, there are

105 - 43 = 62

panels to its right. Removing

24

panels from the right end leaves

62 - 24 = 38

panels to its right. Since there are

38

panels after it, Panel Sigma must be the

39^{\text{th}}

panel from the right end.

Estimated Time:1m 0s

Question 418Question →

A standard six-sided die (where the sum of the numbers on any pair of opposite faces is exactly $7$ ) is placed on a flat table. In its initial position, the number $4$ is on the top face, the number $2$ is on the front face (facing you), and the number $1$ is on the right face.

The die is then rolled $90^\circ$ over its edges along the table in the following sequence:
1. Rolled to the Right (East).
2. Rolled Forward (North, away from you).
3. Rolled to the Right (East).
4. Rolled Backward (South, towards you).

What number will be on the top face of the die in its final position?

6

3

5

1

Show answer & explanation

Answer: 6

Answer

The number 6 will be on the top face.

By carefully tracking the 3D rotation of the die step-by-step, the top face transitions from 4 to 6 (after the first right roll), to 2 (after the forward roll), to 3 (after the second right roll), and finally to 6 (after the backward roll). The sequence of orthogonal rotations guarantees that the number 6 rests on the top face at the end.

Step-by-Step Solution

1

Determine the initial state of all six faces based on standard die rules.

Top = 4, Bottom = 3, Front = 2, Back = 5, Right = 1, Left = 6.

A standard die has opposite faces that sum to 7. This allows us to deduce the hidden faces from the visible ones.

2

Track face positions after the first roll: 90° to the Right (East).

Top becomes 6, Bottom becomes 1, Front remains 2, Back remains 5, Right becomes 4, Left becomes 3.

Rolling Right pivots the die over the right edge; the Left face (6) moves to the Top, while Front and Back remain unchanged.

3

Track face positions after the second roll: 90° Forward (North).

Top becomes 2, Bottom becomes 5, Front becomes 1, Back becomes 6, Right remains 4, Left remains 3.

Rolling Forward pivots over the back edge; the Front face (2) moves to the Top, while Right and Left remain unchanged.

4

Track face positions after the third roll: 90° to the Right (East).

Top becomes 3, Bottom becomes 4, Front remains 1, Back remains 6, Right becomes 2, Left becomes 5.

Rolling Right again moves the new Left face (3) to the Top, while Front and Back remain unchanged.

5

Track face positions after the final roll: 90° Backward (South).

Top becomes 6, Bottom becomes 1, Front becomes 3, Back becomes 4, Right remains 2, Left remains 5.

Rolling Backward pivots over the front edge; the Back face (6) moves to the Top. Thus, the final number on the top face is 6.

Key Concept

Spatial tracking of rolling dice and orientation of opposite faces.

Estimated Time:1m 30s

Question 419Question →

In a processing queue of a mainframe computer, Job Alpha is positioned $28^{\text{th}}$ from the front of the queue, and Job Beta is positioned $34^{\text{th}}$ from the back of the queue. If there are exactly $12$ jobs currently waiting between Job Alpha and Job Beta, what is the minimum possible total number of jobs in this queue?

Show answer & explanation

Answer: 48

Answer

48

To find the minimum possible number of jobs, we assume an overlapping sequence. The formula for the minimum total in a ranking problem is: Total = (Rank from Front) + (Rank from Back) - (Number of items in between) - 2. Substituting the values gives: 28 + 34 - 12 - 2 = 62 - 14 = 48.

Step-by-Step Solution

1

Identify the queue configuration required for a minimum total.

The configuration must be overlapping, meaning Job Beta is ahead of Job Alpha in the queue.

An overlapping configuration results in the smallest possible total number of items because the counted items from both ends overlap.

2

Apply the minimum total formula for overlapping cases.

Total = Rank of Alpha from front + Rank of Beta from back - Jobs between them - 2

This formula deducts the overlapping region (the jobs between them and the two jobs themselves) which were counted twice.

3

Substitute the provided numerical values into the formula.

Total = 28 + 34 - 12 - 2

Job Alpha is 28th, Job Beta is 34th, and there are 12 jobs in between.

4

Calculate the final value.

Total = 62 - 14 = 48

Basic arithmetic yields the minimum possible number of jobs in the queue.

Key Concept

Order and Ranking (Overlapping/Minimum Capacity Queue)

Question 420Question →

A robotics engineering team is programming three distinct automated paint dispensers on an assembly line. They are calibrated to release a burst of paint every $\frac{4}{5}$ , $\frac{8}{15}$ , and $\frac{12}{25}$ of a second, respectively. Let $L$ be the least common multiple (LCM) of these three time intervals, representing the exact time when all three dispensers fire simultaneously. Let $H$ be the highest common factor (HCF) of the intervals, representing the largest fundamental time unit that can perfectly measure all three intervals. What is the value of the ratio $\frac{L}{H}$ ?

90

\frac{1}{90}

\frac{2}{5}

\frac{32}{125}

Show answer & explanation

Answer: $90$

Answer

The value of the ratio

L/H

is 90.

To solve the problem, we first find the LCM and HCF of the fractions

\frac{4}{5}

,

\frac{8}{15}

, and

\frac{12}{25}

. The formula for the LCM of fractions is

\frac{\text{LCM of numerators}}{\text{HCF of denominators}}

. The numerators are 4, 8, and 12, which have an LCM of 24. The denominators are 5, 15, and 25, which have an HCF of 5. Thus,

L = \frac{24}{5}

. The formula for the HCF of fractions is

\frac{\text{HCF of numerators}}{\text{LCM of denominators}}

. The HCF of 4, 8, and 12 is 4. The LCM of 5, 15, and 25 is 75. Thus,

H = \frac{4}{75}

. Finally, dividing

L

by

H

gives

\frac{24}{5} \div \frac{4}{75}

, which equals

\frac{24}{5} \times \frac{75}{4} = 6 \times 15 = 90

.

Step-by-Step Solution

1

Calculate

L

, the LCM of the three fractions.

L = \frac{24}{5}

The LCM of fractions is the LCM of their numerators divided by the HCF of their denominators. LCM(4, 8, 12) = 24 and HCF(5, 15, 25) = 5.

2

Calculate

H

, the HCF of the three fractions.

H = \frac{4}{75}

The HCF of fractions is the HCF of their numerators divided by the LCM of their denominators. HCF(4, 8, 12) = 4 and LCM(5, 15, 25) = 75.

3

Calculate the ratio

\frac{L}{H}

.

\frac{24/5}{4/75} = \frac{24}{5} \times \frac{75}{4} = 6 \times 15 = 90

To divide by a fraction, multiply by its reciprocal. Simplifying the resulting expression yields 90.

Key Concept

Calculating the LCM and HCF of fractional values

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