All practice questions

Question 41Question →

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 42Question →

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 43Question →

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 44Question →

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 45Question →

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 46Question →

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 47Question →

Four navigation buoys in a harbor flash at regular intervals of $3 \frac{1}{3}$ minutes, $4 \frac{1}{6}$ minutes, $5 \frac{5}{9}$ minutes, and $7 \frac{1}{2}$ minutes, respectively. If all four buoys flash simultaneously at exactly 12:00 noon, how many additional times will they all flash together again within the next $12$ hours?

Show answer & explanation

Answer: 4

Answer

4

The correct answer is derived by finding the Least Common Multiple of the fractional intervals to determine the duration between simultaneous flashes (150 minutes). Then, by converting the 12-hour period into 720 minutes, dividing 720 by 150 yields 4.8, indicating the buoys will flash together 4 additional times within that exact timeframe.

Step-by-Step Solution

1

Convert the mixed numbers into improper fractions.

The intervals are

\frac{10}{3}

,

\frac{25}{6}

,

\frac{50}{9}

, and

\frac{15}{2}

minutes.

Mathematical operations involving LCM and HCF of fractions require them to be in proper or improper fractional forms, rather than mixed numbers.

2

Determine the mathematical formula required for simultaneous events.

We must calculate the Least Common Multiple (LCM) of the intervals using the formula:

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

.

Simultaneous cyclic events synchronize at the lowest common multiple of their individual intervals.

3

Calculate the LCM of the numerators.

The numerators are

10, 25, 50

, and

15

. Their LCM is

150

.

150

is the smallest integer divisible by all four numerators (

150 = 10 \times 15 = 25 \times 6 = 50 \times 3 = 15 \times 10

).

4

Calculate the HCF of the denominators.

The denominators are

3, 6, 9

, and

2

. Their HCF is

1

.

The numbers

2

and

3

are coprime (they share no common factors greater than

1

), forcing the overall HCF of the set to be

1

.

5

Calculate the exact time interval between simultaneous flashes.

The LCM of the intervals is

\frac{150}{1} = 150

minutes.

Applying the results from steps 3 and 4 directly into the fractional LCM formula.

6

Determine how many times this 150-minute interval fits into the 12-hour window.

12 \text{ hours} = 720 \text{ minutes}

.

\lfloor \frac{720}{150} \rfloor = \lfloor 4.8 \rfloor = 4

.

Dividing the total available time by the simultaneous interval length gives the exact count of additional flashes.

Key Concept

LCM of fractions in simultaneous cyclic events

Question 48Question →

Evaluate the remainder when the numeric expression $47^{101} + 111^{101} - 25$ is divided by $16$ .

Show answer & explanation

Answer: 5

Answer

5

By applying the properties of modular arithmetic, we determine that

47 \equiv -1 \pmod{16}

and

111 \equiv -1 \pmod{16}

. Raising these to the 101st power yields

-1

and

-1

. Subtracting 25 gives a total of

-27

. To find the positive remainder, we add the next highest multiple of 16 (which is 32) to -27, yielding a final valid positive remainder of 5.

Step-by-Step Solution

1

Find the remainder of the bases when divided by 16.

47 \equiv -1 \pmod{16}

and

111 \equiv -1 \pmod{16}

.

Using negative remainders (since

47 = 16 \times 3 - 1

and

111 = 16 \times 7 - 1

) significantly simplifies the calculation of large powers.

2

Substitute the simplified bases into the original expression.

(47)^{101} + (111)^{101} - 25 \equiv (-1)^{101} + (-1)^{101} - 25 \pmod{16}

.

The properties of modular arithmetic allow replacing bases with their congruent values modulo the divisor.

3

Evaluate the exponents and the constant term.

(-1) + (-1) - 25 = -27

.

An odd power of a negative number remains negative, so

(-1)^{101} = -1

.

4

Find the equivalent positive remainder for -27 modulo 16.

-27 \equiv 5 \pmod{16}

.

Adding multiples of 16 (in this case,

16 \times 2 = 32

) to

-27

yields a positive remainder in the valid range of

0

to

15

. Calculating

-27 + 32 = 5

gives the final remainder.

Key Concept

Modular arithmetic with large powers using negative remainders

Alternative Method

Instead of converting the bases to -1, one could convert them to positive 15. The expression becomes

15^{101} + 15^{101} - 25

. However, evaluating this requires recognizing that

15 \equiv -1 \pmod{16}

to easily compute the large exponents, which ultimately merges back into the primary solution path.

Estimated Time:1m 30s

Question 49Question →

A digital encryption protocol derives a validation key from the mathematical properties of the integer $12600$ . The key is calculated by multiplying the total quantity of its even positive divisors by the total quantity of its odd positive divisors. What is the exact value of this validation key?

Show answer & explanation

Answer: 972

Answer

972

By finding the prime factorization of

12600

as

2^3 \times 3^2 \times 5^2 \times 7^1

, we can separate the divisors into even and odd categories. Odd divisors utilize only the odd prime factors, yielding a total of

(2+1)(2+1)(1+1) = 18

odd divisors. Even divisors require at least one power of

2

, giving

3 \times (2+1)(2+1)(1+1) = 54

even divisors. Following the protocol's rule, their product is

18 \times 54 = 972

.

Step-by-Step Solution

1

Find the prime factorization of

12600

.

12600 = 2^3 \times 3^2 \times 5^2 \times 7^1

Prime factorization is necessary to determine the number and types of divisors an integer has.

2

Calculate the quantity of odd positive divisors.

Number of odd divisors =

18

Odd divisors are formed by using only the odd prime factors (

3

,

5

, and

7

). By adding

1

to each of their exponents and multiplying them, we get

(2+1) \times (2+1) \times (1+1) = 3 \times 3 \times 2 = 18

.

3

Calculate the quantity of even positive divisors.

Number of even divisors =

54

Even divisors must include at least one factor of

2

. The number of choices for the power of

2

is equal to its exponent (

3

choices:

2^1, 2^2, 2^3

). Multiplying this by the choices for odd prime factors gives

3 \times (2+1) \times (2+1) \times (1+1) = 3 \times 3 \times 3 \times 2 = 54

.

4

Multiply the number of even divisors by the number of odd divisors to find the validation key.

54 \times 18 = 972

The encryption protocol specifically defines the key as the product of these two calculated quantities.

Key Concept

Prime Factorization and Number of Divisors (Even and Odd)

Question 50Question →

A large solid cube of side length $6 \text{ cm}$ has its opposite faces painted with the same color. The colors used are red, blue, and yellow. After the paint dries, the large cube is completely cut into smaller cubes of side length $1 \text{ cm}$ each. How many of these smaller cubes will have exactly two faces painted, with one face being red and the other face being yellow?

Show answer & explanation

Answer: 16

Answer

16 cubes

There are exactly 4 edges where a red face and a yellow face meet. Each of these edges contains exactly 4 middle cubes (excluding the corner cubes) that have exactly two faces painted. Therefore, multiplying

4 \text{ edges} \times 4 \text{ cubes}

gives 16.

Step-by-Step Solution

1

Calculate the number of smaller cubes along one edge of the large cube.

n = 6 / 1 = 6

cubes per edge.

The value

n

is required to determine how many cubes lie along any given dimension.

2

Determine the number of edges where a red face and a yellow face intersect.

There are exactly 4 intersecting edges.

The cube has 2 red faces (top/bottom) and 2 yellow faces (left/right). Each of the 2 red faces meets both of the 2 yellow faces, resulting in

2 \times 2 = 4

edges.

3

Calculate how many cubes on a single edge have exactly two faces painted.

n - 2 = 6 - 2 = 4

cubes per edge.

The cubes at the two extreme ends of any edge are corner cubes with three painted faces, so they must be subtracted from the total edge length.

4

Multiply the number of relevant edges by the number of two-face painted cubes per edge.

4 \text{ edges} \times 4 \text{ cubes} = 16

cubes.

This yields the total count of smaller cubes possessing exactly one red and one yellow painted face.

Key Concept

Calculation of painted smaller cubes resulting from cutting a colored larger cube based on intersecting faces.

Question 51Question →

A computer algorithm processes a data block and assigns it a routing ID based on modular arithmetic. This ID corresponds to the strictly positive remainder left after dividing the value of $56^{103} - 45$ by $13$ .

Find the value of this routing ID.

Show answer & explanation

Answer: 11

Answer

11

By reducing each component of the expression modulo 13,

56^{103}

reduces to

4^{103}

, which simplifies to

4

because

4^3 \equiv -1 \pmod{13}

. Subtracting

45 \pmod{13}

, which is

6

, yields

-2

. Adding the divisor 13 to

-2

gives the strictly positive remainder of 11.

Step-by-Step Solution

1

Reduce the base 56 modulo 13.

56 \equiv 4 \pmod{13}

Modular arithmetic allows simplifying the base before exponentiation.

2

Find a pattern to easily evaluate

4^{103} \pmod{13}

.

4^3 = 64 \equiv -1 \pmod{13}

Finding a small power that is congruent to 1 or -1 simplifies large exponents significantly.

3

Evaluate

4^{103} \pmod{13}

using the established pattern.

4^{103} = (4^3)^{34} \times 4^1 \equiv (-1)^{34} \times 4 = 4 \pmod{13}

Exponent rules allow breaking down 103 into

3 \times 34 + 1

.

4

Reduce the subtracted constant 45 modulo 13.

45 = 13 \times 3 + 6 \equiv 6 \pmod{13}

All terms in the expression must be evaluated under the same modulus.

5

Subtract the reduced values.

4 - 6 = -2

Substitute the individual modulo results back into the original expression structure.

6

Convert the negative remainder into a positive remainder.

-2 + 13 = 11

The question asks for a strictly positive remainder, which is standard in modular division contexts.

Key Concept

Modular Arithmetic and Negative Remainders

Question 52Question →

In a technology seminar attended by $60$ software engineers, $35$ engineers know Python and $25$ engineers know Java. If $10$ engineers know both Python and Java, how many engineers know neither of these two programming languages?

Show answer & explanation

Answer: 10

Answer

The number of engineers who know neither Python nor Java is 10.

The correct answer is 10. By taking the sum of Python knowers (

35

) and Java knowers (

25

) and subtracting those who know both (

10

), we obtain

50

engineers who know at least one language. Subtracting this from the total

60

seminar participants leaves

10

engineers who know neither language.

Step-by-Step Solution

1

Find the total number of engineers who know at least one programming language

50 engineers know at least one language

Apply the two-set union formula:

n(A \cup B) = n(A) + n(B) - n(A \cap B) = 35 + 25 - 10 = 50

.

2

Calculate the number of engineers who know neither language

10 engineers know neither language

Subtract the number of engineers knowing at least one language from the total group size:

60 - 50 = 10

.

Key Concept

Two-Set Inclusion-Exclusion Principle and Complement of Sets

Question 53Question →

An archaeological surveyor starting from a central benchmark walks $20\text{ m}$ due East. He then turns $135^\circ$ to his left and walks $14\sqrt{2}\text{ m}$ . Next, he turns $45^\circ$ to his left and walks $18\text{ m}$ . Finally, he turns $90^\circ$ to his right and walks $2\text{ m}$ to reach an excavation site. What is the shortest direct distance (in meters) between the starting benchmark and the excavation site?

Show answer & explanation

Answer: 20

Answer

The shortest direct distance between the starting benchmark and the excavation site is 20 meters.

By resolving each sequential movement into Cartesian coordinate components

(x, y)

, the final position relative to origin

(0,0)

is found to be

(-12\text{ m}, 16\text{ m})

. The straight-line distance is computed using the distance formula

\sqrt{(-12)^2 + (16)^2} = \sqrt{144 + 256} = \sqrt{400} = 20\text{ m}

.

Step-by-Step Solution

1

Plot the initial displacement on a Cartesian coordinate plane with origin

(0,0)

at the starting benchmark.

Position after walking

20\text{ m}

East is

(20, 0)

facing East (

0^\circ

).

East represents the positive x-axis direction.

2

Calculate displacement after turning

135^\circ

left (facing North-West) and walking

14\sqrt{2}\text{ m}

.

Position becomes

(20 - 14, 0 + 14) = (6, 14)

.

In North-West direction, horizontal component is

-14\sqrt{2} \times \frac{1}{\sqrt{2}} = -14\text{ m}

and vertical component is

+14\sqrt{2} \times \frac{1}{\sqrt{2}} = +14\text{ m}

.

3

Calculate displacement after turning

45^\circ

left from North-West (facing West) and walking

18\text{ m}

.

Position becomes

(6 - 18, 14) = (-12, 14)

.

Turning

45^\circ

left from North-West aligns facing direction with West (negative x-axis).

4

Calculate displacement after turning

90^\circ

right from West (facing North) and walking

2\text{ m}

.

Final position is

(-12, 14 + 2) = (-12, 16)

.

Turning

90^\circ

right from West aligns facing direction with North (positive y-axis).

5

Calculate the direct straight-line distance from starting point

(0,0)

to final position

(-12, 16)

.

Distance

D = \sqrt{(-12)^2 + 16^2} = \sqrt{144 + 256} = \sqrt{400} = 20\text{ m}

.

Applying the Pythagorean theorem

D = \sqrt{x^2 + y^2}

yields the shortest distance.

Key Concept

Coordinate Geometry and Vector Resolution in Direction and Distance Problems

Estimated Time:2m 0s

Question 54Question →

Analyze the pattern in the numerical sequence given below:

4, \quad 9, \quad 20, \quad 43, \quad 90, \quad ?

What is the value of the missing term that completes the sequence?

Show answer & explanation

Answer: 185

Answer

185

Each consecutive term is calculated by doubling the previous term and adding an increasing integer (

+1, +2, +3, +4, +5

). Following this rule,

90 \times 2 + 5 = 185

.

Step-by-Step Solution

1

Examine differences or operations between adjacent terms

9 - 4 = 5

,

20 - 9 = 11

,

43 - 20 = 23

,

90 - 43 = 47

Identify the primary numerical increment pattern

2

Analyze the rule for term progression

The operations follow

T_n = T_{n-1} \times 2 + (n-1)

for

n \ge 2

Confirm consistency across all given terms

3

Compute the sixth term (

T_6

)

T_6 = 90 \times 2 + 5 = 185

Apply the addition increment

+5

following

+4

Key Concept

Number and Alphabet Series

Question 55Question →

A wholesale merchant has three batches of specialty syrups measuring $\frac{63}{5}$ liters, $\frac{72}{25}$ liters, and $\frac{81}{10}$ liters in volume. To distribute them in uniform sample sizes, the merchant wants to repackage all the syrup into identical smaller bottles filled exactly to the brim, ensuring no syrup is left over from any of the original batches. What is the maximum possible capacity (in liters) of each sample bottle?

Show answer & explanation

Answer: 0.18

Answer

The maximum possible capacity of each sample bottle is 0.18 liters.

To find the maximum capacity of uniform bottles that can hold the syrup from any batch without leftover, we must find the Highest Common Factor (HCF) of the respective volumes. For fractions, the rule is

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

. The HCF of the numerators (

63, 72, 81

) is

9

, and the LCM of the denominators (

5, 25, 10

) is

50

. The resulting maximum capacity is

\frac{9}{50}

, which equals

0.18

liters.

Step-by-Step Solution

1

Determine the mathematical operation required.

Calculate the HCF of

\frac{63}{5}

,

\frac{72}{25}

, and

\frac{81}{10}

.

Finding the largest equal container size that leaves no remainder requires computing the Highest Common Factor.

2

Apply the rule for finding the HCF of fractions.

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

This standard formula correctly identifies the greatest common divisor for a set of rational numbers.

3

Find the HCF of the numerators.

\text{HCF}(63, 72, 81) = 9

The largest integer that perfectly divides 63, 72, and 81 is 9.

4

Find the LCM of the denominators.

\text{LCM}(5, 25, 10) = 50

The smallest integer that is a multiple of 5, 25, and 10 is 50.

5

Compute the final fraction and convert it to a decimal.

\frac{9}{50} = 0.18

Dividing the HCF of the numerators by the LCM of the denominators yields the exact capacity in liters.

Key Concept

HCF and LCM of fractions

Question 56Question →

A local water reservoir was initially filled to exactly $0.\overline{3}$ of its maximum capacity. After several days of heavy rainfall, an additional $14.4$ million liters of water flowed into the reservoir, bringing the water level to $0.6$ of its maximum capacity.

Currently, a nearby town consumes water from the reservoir at a steady rate of $1\frac{1}{2}$ million liters per day. Additionally, an undiscovered leak at the base of the reservoir drains water at a constant rate of $0.05$ million liters per hour.

If the reservoir were completely full and no further water was added, in how many days would it become completely empty?

Show answer & explanation

Answer: 20

Answer

20

The total capacity of the reservoir is found by equating the fractional increase (

\frac{3}{5} - \frac{1}{3} = \frac{4}{15}

) to the volume added (

14.4

million liters), yielding

54

million liters. The total daily water depletion is the sum of the town's daily usage (

1.5

million liters) and the leak converted to a daily rate (

0.05 \times 24 = 1.2

million liters), giving

2.7

million liters per day. Dividing

54

by

2.7

gives exactly

20

days.

Step-by-Step Solution

1

Convert the decimal fill levels into fractions and compute their difference.

Initial level =

\frac{1}{3}

. Final level =

\frac{3}{5}

. Difference =

\frac{3}{5} - \frac{1}{3} = \frac{4}{15}

.

Working with exact fractions avoids rounding errors from recurring decimals and simplifies finding the proportion of water added.

2

Calculate the maximum capacity of the reservoir.

\frac{4}{15} \times \text{Capacity} = 14.4 \implies \text{Capacity} = 14.4 \times \frac{15}{4} = 54

million liters.

The difference in the fractional water level represents exactly the volume of rain added.

3

Calculate the total volume of water lost per day.

Town =

1.5

million L/day. Leak =

0.05 \times 24 = 1.2

million L/day. Total =

2.7

million L/day.

The rates must be in the same time unit (days) before they can be accurately combined.

4

Determine the time required to empty the completely full reservoir.

54 \div 2.7 = 20

days.

Dividing the total capacity by the combined daily rate of depletion gives the time in days.

Key Concept

Fractions and Decimals

Estimated Time:3m 0s

Question 57Question →

In a regional agricultural department, a survey was conducted among $160$ field officers regarding their supervisory duties across three major crop initiatives: Wheat ( $W$ ), Rice ( $R$ ), and Cotton ( $C$ ). The survey revealed that $85$ officers supervise Wheat, $70$ supervise Rice, and $60$ supervise Cotton. Furthermore, $35$ officers supervise both Wheat and Rice, $25$ supervise both Rice and Cotton, $30$ supervise both Wheat and Cotton, and $15$ officers supervise all three crop initiatives. How many field officers supervise exactly two of these three crop initiatives?

Show answer & explanation

Answer: 45

Answer

The total number of field officers who supervise exactly two crop initiatives is 45.

To find the number of officers supervising exactly two crop initiatives, we calculate the number of officers in each pair of overlapping sets exclusive of the three-set intersection: for Wheat and Rice only (

35 - 15 = 20

), for Rice and Cotton only (

25 - 15 = 10

), and for Wheat and Cotton only (

30 - 15 = 15

). Summing these three regions gives

20 + 10 + 15 = 45

.

Step-by-Step Solution

1

Extract intersection values for each pair of sets and subtract the central intersection of all three sets

Only Wheat and Rice =

35 - 15 = 20

; Only Rice and Cotton =

25 - 15 = 10

; Only Wheat and Cotton =

30 - 15 = 15

The given pairwise intersections include officers who supervise all three initiatives. To isolate those who supervise exactly two, the three-set intersection must be removed from each pairwise intersection.

2

Sum the three distinct 'exactly two' regions

Total =

20 + 10 + 15 = 45

Adding these disjoint counts yields the total number of officers supervising precisely two crop initiatives.

Key Concept

Inclusion-Exclusion Principle and Venn Diagram Region Analysis for Three Overlapping Sets

Question 58Question →

In a class merit list, Rahul is ranked $15^{\text{th}}$ from the top and $21^{\text{st}}$ from the bottom. What is the total number of students in the class?

Show answer & explanation

Answer: 35

Answer

The total number of students in the class is

35

.

When the position of a single person is given from both ends of a line, the total count is calculated using the formula

\text{Total} = \text{Position from Left/Top} + \text{Position from Right/Bottom} - 1

. Substituting the values gives

15 + 21 - 1 = 35

.

Step-by-Step Solution

1

Identify the given ranks from both ends for the same person.

Rank from top = 15, Rank from bottom = 21.

The position of Rahul is provided relative to both the start and end of the queue.

2

Apply the single-person dual rank total calculation formula.

\text{Total} = \text{Rank}_{\text{top}} + \text{Rank}_{\text{bottom}} - 1

Since Rahul is included in both rank counts, adding the ranks counts him twice, requiring a subtraction of 1.

3

Perform the computation.

15 + 21 - 1 = 35

Simplifying the arithmetic yields the exact total headcount of the class.

Key Concept

Single-Person Dual Rank Total Calculation

Question 59Question →

In a district administration department, a survey was conducted among $250$ officers regarding their technical expertise in three digital governance domains: Cyber Security ( $C$ ), Data Analytics ( $D$ ), and Public Grievance Portals ( $P$ ). The data collected is as follows:
- $120$ officers have expertise in Data Analytics.
- $115$ officers have expertise in Cyber Security.
- $100$ officers have expertise in Public Grievance Portals.
- $50$ officers have expertise in both Data Analytics and Cyber Security.
- $45$ officers have expertise in both Cyber Security and Public Grievance Portals.
- $40$ officers have expertise in both Data Analytics and Public Grievance Portals.
- $20$ officers possess expertise in all three digital governance domains.

Based on the given data, how many officers have expertise in EXACTLY TWO of these three domains?

Show answer & explanation

Answer: 75

Answer

The total number of officers who have expertise in exactly two digital governance domains is 75.

To find the number of officers with expertise in exactly two domains, we calculate the exclusive intersection regions by subtracting the central three-set intersection (

20

) from each of the given double-intersection totals:

(50 - 20) + (45 - 20) + (40 - 20) = 30 + 25 + 20 = 75

.

Step-by-Step Solution

1

Calculate the number of officers in the region representing expertise in ONLY Data Analytics and Cyber Security

50 - 20 = 30

officers

The total intersection of Data Analytics and Cyber Security includes officers proficient in all three domains. Subtracting the 3-domain count isolates those proficient in only these two.

2

Calculate the number of officers in the region representing expertise in ONLY Cyber Security and Public Grievance Portals

45 - 20 = 25

officers

Subtracting the 3-domain count from the total intersection of Cyber Security and Public Grievance Portals isolates those proficient in only these two.

3

Calculate the number of officers in the region representing expertise in ONLY Data Analytics and Public Grievance Portals

40 - 20 = 20

officers

Subtracting the 3-domain count from the total intersection of Data Analytics and Public Grievance Portals isolates those proficient in only these two.

4

Sum the three mutually exclusive regions representing expertise in exactly two domains

30 + 25 + 20 = 75

officers

Adding these disjoint sets yields the total number of officers with expertise in exactly two of the three digital governance domains.

Key Concept

Principle of Inclusion-Exclusion and Venn Diagram Region Partitioning

Estimated Time:2m 0s

Question 60Question →

In a published civil services merit list, Rohan is ranked $14^{\text{th}}$ from the top, while Seema is ranked $33^{\text{rd}}$ from the bottom. After a re-evaluation of marks, Rohan and Seema interchange their positions, resulting in Rohan becoming $48^{\text{th}}$ from the top. A third candidate, Amit, holds a rank exactly halfway between Rohan's new position and Seema's new position. If $6$ candidates ranked above Amit withdraw their candidature and $4$ new candidates are placed below Amit in the updated list, what is Amit's final rank from the bottom of the modified merit list?

Show answer & explanation

Answer: 54

Answer

Amit's final rank from the bottom of the modified merit list is 54.

Seema's dual information (

48^{\text{th}}

from top and

33^{\text{rd}}

from bottom) gives an initial total of 80 candidates. Amit's midpoint position between

14^{\text{th}}

and

48^{\text{th}}

is

31^{\text{st}}

from top, placing 49 candidates below him. Accounting for 6 withdrawals above and 4 additions below leaves 53 candidates below Amit, establishing his final rank as

54^{\text{th}}

from the bottom.

Step-by-Step Solution

1

Find total candidates in the original merit list using the interchange data.

Total candidates N = 80

When Rohan and Seema interchange positions, Rohan occupies Seema's original spot (

48^{\text{th}}

from top). Since Seema was

33^{\text{rd}}

from bottom,

N = 48 + 33 - 1 = 80

.

2

Calculate Amit's initial rank from the top.

Amit's initial top rank = 31st

Amit is positioned halfway between Seema's new position (

14^{\text{th}}

from top) and Rohan's new position (

48^{\text{th}}

from top), so rank =

(14 + 48) / 2 = 31^{\text{st}}

.

3

Find initial counts of candidates ranked above and below Amit.

30 candidates above, 49 candidates below

With a top rank of

31^{\text{st}}

out of 80, there are

31 - 1 = 30

candidates above Amit and

80 - 31 = 49

candidates below Amit.

4

Adjust counts based on candidate withdrawals and additions.

24 candidates above, 53 candidates below

Subtract 6 from candidates above (

30 - 6 = 24

) and add 4 to candidates below (

49 + 4 = 53

).

5

Calculate Amit's revised rank from the bottom.

54th from the bottom

The rank from the bottom equals (candidates below + 1), which gives

53 + 1 = 54

.

Key Concept

Dual Rank Position Interchange, Midpoint Computation, and Dynamic List Adjustment

Estimated Time:2m 30s

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