All practice questions

Question 601Question →

A synchronized scheduling system operates on a repeating 19-millisecond cycle. A specific event is triggered at a timestamp $T$ in milliseconds, given by the formula $T = 37^{45} - 17 \times 40^{23}$ . To find the exact position within the current cycle when the event occurs, the system calculates the positive remainder when $T$ is divided by $19$ . At what millisecond mark within the cycle does the event trigger?

Show answer & explanation

Answer: 6

Answer

6

By applying modular arithmetic rules, Fermat's Little Theorem reduces the large exponents. The first term evaluates to

-1 \pmod{19}

and the second term evaluates to

12 \pmod{19}

. Their difference is

-13 \pmod{19}

, which corresponds to a positive remainder of

6

.

Step-by-Step Solution

1

Simplify the first term

37^{45}

modulo 19.

37^{45} \equiv -1 \pmod{19}

Since

37 = 19 \times 2 - 1

, it follows that

37 \equiv -1 \pmod{19}

. An odd power of

-1

is

-1

.

2

Simplify the base of the second term,

40^{23}

, modulo 19.

40^{23} \equiv 2^{23} \pmod{19}

Because

40 = 19 \times 2 + 2

, we can replace the base

40

with its remainder

2

.

3

Use Fermat's Little Theorem to reduce the exponent in

2^{23} \pmod{19}

.

2^{23} \equiv 13 \pmod{19}

Fermat's Little Theorem states

a^{p-1} \equiv 1 \pmod{p}

for a prime

p

. Here,

2^{18} \equiv 1 \pmod{19}

. Therefore,

2^{23} = 2^{18} \times 2^5 \equiv 1 \times 32 \equiv 13 \pmod{19}

.

4

Multiply by 17 and find the remainder of the second term.

17 \times 40^{23} \equiv 12 \pmod{19}

We can write

17 \equiv -2 \pmod{19}

. Then,

(-2) \times 13 = -26

. Adding a multiple of 19 (which is 38) gives

-26 + 38 = 12

.

5

Subtract the second term from the first and convert to a positive remainder.

The final remainder is

6

.

-1 - 12 = -13

. To get the positive remainder, add the modulus 19:

-13 + 19 = 6

.

Key Concept

Applying modular arithmetic rules, properties of negative remainders, and Fermat's Little Theorem to simplify large exponential expressions.

Estimated Time:1m 30s

Question 602Question →

An artisan is preparing metal rods for a custom fence. Two existing rods, which measure $\frac{15}{4}$ meters and $\frac{25}{8}$ meters in length, must be cut into identical smaller pieces of the maximum possible length without wasting any material. What should be the length of each piece?

\frac{75}{4}

meters

\frac{5}{4}

meters

\frac{5}{8}

meters

\frac{75}{8}

meters

Show answer & explanation

Answer: $\frac{5}{8}$ meters

Answer

\frac{5}{8}

meters

The maximum possible length of each piece is found by calculating the HCF of the two fractional lengths. The correct formula is to divide the HCF of the numerators (15 and 25) by the LCM of the denominators (4 and 8). The HCF of 15 and 25 is 5, and the LCM of 4 and 8 is 8, resulting in exactly

\frac{5}{8}

meters.

Step-by-Step Solution

1

Identify the mathematical operation required for the scenario.

Finding the maximum possible identical length from two given lengths requires calculating their Highest Common Factor (HCF).

The pieces must be of equal length and as large as possible without leaving any remainder.

2

State the formula for finding the HCF of fractions.

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

This is the standard rule for determining the greatest common divisor of rational numbers.

3

Calculate the HCF of the numerators (15 and 25).

The factors of 15 are 1, 3, 5, 15. The factors of 25 are 1, 5, 25. The highest common factor is 5.

The numerator of our final answer must be the HCF of the original numerators.

4

Calculate the LCM of the denominators (4 and 8).

The multiples of 8 (8, 16, 24...) are already divisible by 4. Thus, the least common multiple is 8.

The denominator of our final answer must be the LCM of the original denominators.

5

Construct the final fraction.

\frac{5}{8}

meters.

Combining the results from the previous steps yields the correct maximum length.

Key Concept

Highest Common Factor (HCF) of fractions

Question 603Question →

Consider the algebraic fraction below:

\frac{5^{n+2} - 5^n}{5^{n+1} - 5^n}

Which of the following represents the simplified exact value of this expression?

4

5

6

25

Show answer & explanation

Answer: 6

Answer

6

The expression can be systematically simplified by factoring out the lowest common power of the base, which is

5^n

, from both the numerator and the denominator. The numerator becomes

5^n(5^2 - 1) = 5^n(24)

. The denominator becomes

5^n(5^1 - 1) = 5^n(4)

. Canceling the common factor

5^n

from the top and bottom leaves

24 / 4

, which equals 6.

Step-by-Step Solution

1

Identify the greatest common factor in both the numerator and the denominator.

The common factor is

5^n

.

Factoring out the lowest power of the base simplifies algebraic expressions involving exponential terms.

2

Factor out

5^n

from the numerator.

5^{n+2} - 5^n = 5^n(5^2 - 1)

Applying the exponent rule

a^{x+y} = a^x \times a^y

in reverse allows us to extract the common term.

3

Factor out

5^n

from the denominator.

5^{n+1} - 5^n = 5^n(5^1 - 1)

Applying the same extraction process to the bottom expression.

4

Substitute the factored forms back into the fraction and cancel the common

5^n

term.

\frac{5^n(25 - 1)}{5^n(5 - 1)} = \frac{24}{4}

The

5^n

multiplier in the numerator and denominator cancel each other out, leaving basic arithmetic values.

5

Perform the final arithmetic division.

24 \div 4 = 6

Simplifying the remaining numerical fraction yields the final exact value.

Key Concept

Simplifying exponential expressions by factoring out common base powers.

Estimated Time:1m 15s

Question 604Question →

Eight officials (Auditor, Clerk, Governor, Judge, Mayor, Senator, Sheriff, and Treasurer) are seated around a square table. Four of them sit at the corners facing the center of the table, while the other four sit in the middle of each side facing outward away from the center.

- The Sheriff sits at one of the corners.
- The Governor sits third to the right of the Sheriff.
- The Mayor sits second to the right of the Governor.
- The Senator and the Judge are immediate neighbors, but neither is an immediate neighbor of the Governor or the Sheriff.
- The Treasurer sits third to the right of the Judge.
- The Clerk is not an immediate neighbor of the Governor.

Based on the given seating arrangement, which of the following statements are correct?

Select all that apply

The Clerk and the Governor sit exactly opposite each other.

The Auditor sits second to the left of the Treasurer.

Exactly three people sit between the Judge and the Mayor.

The Senator sits to the immediate left of the Clerk.

Show answer & explanation

Answer: The Clerk and the Governor sit exactly opposite each other.; Exactly three people sit between the Judge and the Mayor.

Answer

The statements indicating that the Clerk and Governor sit opposite each other, and that exactly three people sit between the Judge and the Mayor, are correct.

Based on the systematic deduction of all constraints, the table is arranged as follows (clockwise from top-left): Sheriff, Clerk, Senator, Judge, Auditor, Governor, Treasurer, and Mayor. In this specific configuration, the Clerk and Governor occupy directly opposite middle seats, and the Judge and Mayor occupy opposite middle seats, meaning there are exactly three people between them on either side.

Step-by-Step Solution

1

Establish the directional rules for the inward and outward facing seats.

For the 4 corners facing inward, 'right' moves counter-clockwise and 'left' moves clockwise. For the 4 middle positions facing outward, 'right' moves clockwise and 'left' moves counter-clockwise.

Correctly identifying left and right based on facing direction is mandatory before placing any individuals.

2

Place the Sheriff, Governor, and Mayor to create a foundational layout.

Place the Sheriff at a corner (e.g., top-left, inward). Third to the right (counter-clockwise) places the Governor at the bottom-middle position (outward). Second to the right of the Governor (clockwise) places the Mayor at the left-middle position (outward).

These three linked conditions provide fixed anchor points on the table.

3

Determine the available positions for the Senator and the Judge.

They must sit in adjacent seats but cannot sit next to the Governor or the Sheriff. Eliminating the seats adjacent to the Governor and Sheriff leaves only the top-right corner and the right-middle position as the single valid pair of adjacent seats.

Negative constraints drastically reduce the possible locations for grouped individuals.

4

Differentiate the specific seats for the Senator and the Judge using the Treasurer's condition.

The Treasurer is third to the right of the Judge. If the Judge were at the top-right corner (inward), third to the right would be the left-middle seat, which is already occupied by the Mayor. Thus, the Judge must be at the right-middle seat (outward). Third to the right (clockwise) places the Treasurer at the bottom-left corner. The Senator takes the top-right corner.

Testing the directional constraint ensures no two people are assigned to the same seat.

5

Place the remaining officials, the Clerk and the Auditor.

The remaining empty seats are the top-middle and bottom-right corner. The Clerk cannot be a neighbor of the Governor, so the Clerk takes the top-middle seat. The Auditor fills the final seat at the bottom-right corner.

Applying the final negative constraint finalizes the complete seating arrangement.

Key Concept

Solving complex seating arrangements with mixed facing directions by systematically tracking relative left/right movements.

Question 605Question →

Consider the set of the first 100 positive integers (from 1 to 100 inclusive).

An integer $N$ from this set satisfies all of the following three conditions simultaneously:
1. $N$ is a composite number.
2. $N$ is neither divisible by 2 nor divisible by 3.
3. The square root of $N$ is an irrational number.

What is the total number of possible values for $N$ ?

Show answer & explanation

Answer: 7

Answer

There are exactly 7 values for N that satisfy all three conditions.

The complete set of numbers not divisible by 2 or 3 from 1 to 100 contains 33 integers. Excluding the number 1 (neither prime nor composite) and the 23 prime numbers leaves exactly 9 composite numbers. From these 9 composites, 25 and 49 must be excluded because their square roots are 5 and 7 (rational numbers). This leaves 7 valid integers.

Step-by-Step Solution

1

Determine the total number of integers from 1 to 100 that are neither divisible by 2 nor divisible by 3.

There are 33 such numbers.

Using the inclusion-exclusion principle: there are 50 multiples of 2, 33 multiples of 3, and 16 multiples of 6. Multiples of 2 or 3 = 50 + 33 - 16 = 67. The remaining numbers are 100 - 67 = 33.

2

Filter the 33 remaining integers to find those that are composite.

There are 9 composite numbers: 25, 35, 49, 55, 65, 77, 85, 91, and 95.

Of the 33 numbers, 1 is neither prime nor composite. There are 25 primes up to 100, and excluding 2 and 3 leaves 23 primes. Thus, the composites are 33 - 1 - 23 = 9. These are the products of primes 5 and greater.

3

Eliminate numbers from the composite list whose square roots are rational.

Remove 25 and 49.

Condition 3 requires the square root of N to be irrational. An integer has a rational square root if and only if it is a perfect square. In our list, 25 and 49 are perfect squares.

4

Count the final remaining valid numbers.

7 numbers remain.

Subtracting the 2 perfect squares from the 9 composite numbers leaves 7 numbers that satisfy all three conditions.

Key Concept

Classification of numbers combining prime/composite definitions, divisibility principles, and properties of rational and irrational roots.

Question 606Question →

A logistics manager is packing identical relief kits into crates. When she attempts to pack them in equal batches of $16$ , $24$ , $30$ , or $36$ kits per crate, there are always exactly $8$ kits left over. However, if she packs them in batches of exactly $19$ kits per crate, there are zero kits left over. What is the least possible total number of relief kits she could be packing?

Show answer & explanation

Answer: 2888

Answer

The least possible total number of relief kits is 2888.

The correct answer is derived by finding the general form of a number that leaves a remainder of 8 when divided by 16, 24, 30, and 36. This form is

720k + 8

. By applying the final condition that the total number must be exactly divisible by 19, we find the smallest valid multiplier is

k=4

. Substituting this back gives

720(4) + 8 = 2888

.

Step-by-Step Solution

1

Establish the relationship for the total number of kits based on the remainders.

The number of kits,

N

, leaves a remainder of

8

when divided by

16

,

24

,

30

, and

36

. Thus,

N = \text{LCM}(16, 24, 30, 36) \times k + 8

.

Any number that leaves the same remainder when divided by multiple divisors can be expressed as a multiple of their least common multiple plus that remainder.

2

Calculate the least common multiple (LCM) of

16

,

24

,

30

, and

36

.

The prime factorizations are

16 = 2^4

,

24 = 2^3 \times 3

,

30 = 2 \times 3 \times 5

, and

36 = 2^2 \times 3^2

. The LCM is the product of the highest powers:

2^4 \times 3^2 \times 5 = 720

.

The LCM is required to find the base repeating cycle for the division condition.

3

Formulate the exact divisibility condition.

Substitute the LCM into the equation to get

N = 720k + 8

. The problem states

N

is exactly divisible by

19

, so

(720k + 8) \equiv 0 \pmod{19}

.

This applies the second constraint of the problem to find the specific multiplier

k

.

4

Simplify the modular arithmetic equation to solve for

k

.

Divide

720

by

19

to find the remainder:

720 = 19 \times 37 + 17

. Substitute

17

for

720

to get

(17k + 8) \equiv 0 \pmod{19}

. This can be written as

(-2k + 8) \equiv 0 \pmod{19}

.

Simplifying large numbers using modulo properties makes finding the integer

k

manageable.

5

Find the smallest positive integer

k

that satisfies the equation.

Solving

-2k + 8 = 0

yields

2k = 8

, which means

k = 4

. Checking:

17(4) + 8 = 68 + 8 = 76

, and

76 \div 19 = 4

, which is exactly divisible.

Finding the smallest valid

k

ensures we calculate the least possible total number of kits.

6

Calculate the final total number of kits.

N = 720(4) + 8 = 2880 + 8 = 2888

.

Substituting

k = 4

back into the original formula for

N

provides the final numerical answer.

Key Concept

Finding a specific numerical value based on multiple simultaneous divisibility and remainder conditions using Least Common Multiple (LCM) and modular arithmetic.

Question 607Question →

Identify all two-digit prime numbers where both the tens digit and the units digit are strictly prime numbers. What is the sum of the largest and the smallest numbers that meet this criterion?

Show answer & explanation

Answer: 96

Answer

96

The correct answer requires finding the intersection of two distinct sets: two-digit numbers formed entirely by prime digits, and two-digit numbers that are mathematically prime. This resulting set is {23, 37, 53, 73}. Adding the minimum value (23) and maximum value (73) yields 96.

Step-by-Step Solution

1

Identify valid single digits.

The single-digit primes available for use are 2, 3, 5, and 7.

The problem states that both individual digits of the target number must be prime.

2

Determine valid units digits.

The units digit can only be 3 or 7.

If the units digit is 2, the number is even. If the units digit is 5, the number is a multiple of 5. Both cases result in a composite two-digit number.

3

List all potential combinations.

The possible combinations are 23, 33, 53, 73, 27, 37, 57, and 77.

These are generated by pairing any prime tens digit {2, 3, 5, 7} with the valid prime units digits {3, 7}.

4

Eliminate composite numbers from the list.

The valid primes are 23, 37, 53, and 73. The numbers 27, 33, 57, and 77 are removed.

27, 33, and 57 are divisible by 3 (sum of digits is a multiple of 3). 77 is divisible by 7.

5

Calculate the final sum.

23 + 73 = 96.

The question asks for the sum of the smallest valid number (23) and the largest valid number (73).

Key Concept

Classification of prime digits and prime numbers.

Question 608Question →

Six journalists—Arthur, Beatrice, Cedric, Daphne, Elias, and Fiona—are seated along one side of a long rectangular press table, all facing the speaker at the front (North). Read the following seating conditions carefully:

1. Arthur is seated third from the left end of the table.
2. Cedric is seated immediately to the right of Arthur.
3. Only one person sits between Cedric and Fiona.
4. Elias sits second to the left of Cedric.
5. Beatrice is not an immediate neighbor of Elias.

Arrange the journalists in their correct seating order from the extreme left end to the extreme right end.

Drag items to arrange them in the correct order

Show answer & explanation

Answer

The correct order from left to right is Daphne, Elias, Arthur, Cedric, Beatrice, and Fiona.

By following the constraints sequentially, each seat resolves uniquely without any contradiction. Arthur's exact position anchors the arrangement, Cedric and Elias fill in relative to him, and Fiona's distance from Cedric forces her to the end. The negative constraint on Beatrice perfectly resolves the final two seats.

Step-by-Step Solution

1

Identify the total number of positions and place Arthur.

Positions: _ _ Arthur _ _ _

The table has 6 seats. The first condition explicitly places Arthur third from the left.

2

Place Cedric relative to Arthur.

Positions: _ _ Arthur Cedric _ _

The second condition states Cedric sits immediately to the right of Arthur, taking the fourth seat.

3

Place Elias relative to Cedric.

Positions: _ Elias Arthur Cedric _ _

The fourth condition states Elias is second to the left of Cedric. Since Cedric is fourth, Elias must be second.

4

Determine Fiona's position.

Positions: _ Elias Arthur Cedric _ Fiona

The third condition states only one person sits between Cedric and Fiona. This means Fiona is either in seat 2 or 6. Since Elias is already in seat 2, Fiona must take seat 6.

5

Use the negative constraint to place Beatrice.

Positions: _ Elias Arthur Cedric Beatrice Fiona

The fifth condition states Beatrice is not an immediate neighbor of Elias. The empty seats are 1 and 5. Seat 1 is next to Elias, so Beatrice must sit in seat 5.

6

Place the final person, Daphne.

Positions: Daphne Elias Arthur Cedric Beatrice Fiona

Daphne is the only remaining journalist and takes the final empty seat at the extreme left (seat 1).

Key Concept

Linear Seating Arrangement with Positional and Negative Constraints

Question 609Question →

A botanist is preparing nutrient solutions. She has three different liquid nutrient extracts measuring $\frac{24}{5}$ liters, $\frac{32}{3}$ liters, and $\frac{40}{7}$ liters. She wants to distribute them entirely into identical small sample vials of maximum possible capacity, such that no extract is left over and the extracts are not mixed. What should be the maximum capacity of each sample vial?

\frac{8}{105}

liters

\frac{1}{480}

liters

480

liters

8

liters

Show answer & explanation

Answer: $\frac{8}{105}$ liters

Answer

The correct maximum capacity is

\frac{8}{105}

liters.

To find the maximum identical capacity that can exactly measure out each of the given quantities without mixing, we must calculate the Highest Common Factor (HCF) of the given fractions. By applying the formula for the HCF of fractions, we divide the HCF of the numerators (8) by the Least Common Multiple (LCM) of the denominators (105), resulting in

\frac{8}{105}

liters.

Step-by-Step Solution

1

Identify the mathematical operation required.

We need to find the Highest Common Factor (HCF) of the three fractions to determine the maximum equal capacity.

The vials must have the maximum possible capacity and perfectly measure out all three volumes without mixing them or leaving any remainder.

2

Recall the formula for the HCF of fractions.

HCF of fractions = (HCF of numerators) / (LCM of denominators).

This is the standard algebraic rule for determining the greatest common divisor of fractional values.

3

Calculate the HCF of the numerators: 24, 32, and 40.

The prime factorizations are

24 = 2^3 \times 3

,

32 = 2^5

, and

40 = 2^3 \times 5

. The highest common factor is

2^3 = 8

.

The HCF of the numerators forms the numerator of our final answer.

4

Calculate the LCM of the denominators: 5, 3, and 7.

Since 5, 3, and 7 are all prime numbers, their LCM is their product:

5 \times 3 \times 7 = 105

.

The LCM of the denominators forms the denominator of our final answer.

5

Construct the final fraction.

The capacity is

\frac{8}{105}

liters.

Dividing the HCF of the numerators by the LCM of the denominators gives the final HCF of the original fractions.

Key Concept

HCF and LCM of fractions

Question 610Question →

Consider the following statements regarding the classification of numbers:

I. The number $0$ is an even integer, but it is classified as neither positive nor negative.
II. The fraction $\frac{22}{7}$ is an irrational number because it is commonly used as the value of the constant $\pi$ .
III. The product of any two distinct irrational numbers is always an irrational number.

Which of the statements given above is/are correct?

I and II only

I and III only

I only

II and III only

Show answer & explanation

Answer: I only

Answer

Only the first statement is correct.

Only the first statement is true. Zero is an even integer because it is an integer multiple of

2

(

0 = 2 \times 0

), and it sits at the origin of the number line, meaning it is neither positive nor negative. The second statement is false because any number that can be expressed as the quotient of two integers, such as

\frac{22}{7}

, is rational by definition. The third statement is false because the product of distinct irrational numbers can be rational (for instance,

\sqrt{2} \times 8\sqrt{2} = 16

).

Step-by-Step Solution

1

Evaluate the parity and sign classification of zero in Statement I.

Zero is divisible by

2

without a remainder (

0 = 2 \times 0

), making it an even integer. It sits exactly between the negative and positive numbers on the number line, so it is strictly neither positive nor negative. Statement I is true.

To verify fundamental integer properties regarding the origin point.

2

Analyze the definition of the fraction in Statement II.

A rational number is any number that can be expressed as a ratio of two integers (

\frac{p}{q}

, where

q \neq 0

). Since

22

and

7

are integers,

\frac{22}{7}

is a rational number. It is merely an approximation of

\pi

, not its exact value. Statement II is false.

To test the distinction between a rational approximation and an irrational constant.

3

Test the closure property of irrational numbers under multiplication for Statement III.

Multiplying two distinct irrational numbers, such as

(2 - \sqrt{3})

and

(2 + \sqrt{3})

, yields

4 - 3 = 1

. Alternatively,

\sqrt{2} \times \sqrt{8} = \sqrt{16} = 4

. Since

1

and

4

are rational numbers, the product of distinct irrationals is not always irrational. Statement III is false.

To determine if irrational numbers are closed under multiplication.

Key Concept

Classification of Real Numbers: Parity of Zero, Rational vs Irrational Definitions, and Closure Properties

Estimated Time:1m 15s

Question 611Question →

Evaluate the following statements regarding the properties and classification of numbers. Which of the statements are mathematically correct? (Select all that apply)

Select all that apply

The product of the irrational numbers

\sqrt{8}

and

\sqrt{18}

is a composite integer.

The unit digit of the exponential number

2^{24}

is an odd integer.

The sum of any two distinct odd prime numbers is always an even composite integer.

The remainder obtained when

-17

is divided by

5

is an even integer.

Show answer & explanation

Answer: The product of the irrational numbers $\sqrt{8}$ and $\sqrt{18}$ is a composite integer.; The sum of any two distinct odd prime numbers is always an even composite integer.

Answer

The statement regarding the product of

\sqrt{8}

and

\sqrt{18}

being a composite integer, and the statement that the sum of any two distinct odd prime numbers is an even composite integer, are both mathematically correct.

The statement about the irrational product is correct because

\sqrt{8} \times \sqrt{18} = 12

, and 12 is a composite integer. The statement regarding prime numbers is correct because adding any two odd numbers always yields an even number. Since the smallest distinct odd primes are 3 and 5 (summing to 8), their sum will always be an even integer strictly greater than 2, which guarantees it is composite.

Step-by-Step Solution

1

Evaluate the product of the given irrational numbers.

\sqrt{8} \times \sqrt{18} = \sqrt{144} = 12

. Since 12 has factors other than 1 and itself, it is a composite integer.

To verify if the statement about the irrational product correctly classifies the resulting number.

2

Calculate the unit digit of

2^{24}

.

The exponent 24 is perfectly divisible by 4 (

24 \pmod 4 = 0

). We use the 4th power in the cycle:

2^4 = 16

. The unit digit is 6, which is even.

To determine the parity of the unit digit for the relevant statement.

3

Analyze the sum of two distinct odd prime numbers.

An odd integer added to another odd integer yields an even integer. The smallest possible sum is

3 + 5 = 8

. Any even integer greater than 2 is composite.

To mathematically prove the validity of the statement concerning prime sums.

4

Determine the remainder when

-17

is divided by 5.

Using modulo arithmetic,

-17 = 5 \times (-4) + 3

. The true positive remainder is 3, which is an odd integer.

To correctly evaluate the division operation and classify the remainder for the modulo statement.

Key Concept

Classification of Numbers and Number Properties

Estimated Time:1m 30s

Question 612Question →

Calculate the lowest common multiple (LCM) of the fractions $\frac{3}{4}$ , $\frac{9}{10}$ , and $\frac{15}{16}$ . Express your final answer as a decimal.

Show answer & explanation

Answer: 22.5

Answer

22.5

The lowest common multiple (LCM) of a set of fractions is found by dividing the LCM of their numerators by the highest common factor (HCF) of their denominators. For the numerators

3

,

9

, and

15

, the LCM is

45

. For the denominators

4

,

10

, and

16

, the HCF is

2

. Thus, the LCM of the fractions is

\frac{45}{2}

, which equals

22.5

in decimal form.

Step-by-Step Solution

1

Recall the mathematical formula for finding the LCM of multiple fractions.

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

This formula is the standard method to determine the lowest common multiple when dealing with fractional values.

2

Identify the numerators and calculate their lowest common multiple (LCM).

The numerators are

3

,

9

, and

15

. Their LCM is

45

(since

45

is the smallest number perfectly divisible by

3

,

9

, and

15

).

The numerator of the final fraction requires the LCM of all the given numerators.

3

Identify the denominators and calculate their highest common factor (HCF).

The denominators are

4

,

10

, and

16

. Their HCF is

2

(since

2

is the largest number that divides

4

,

10

, and

16

without a remainder).

The denominator of the final fraction requires the HCF of all the given denominators.

4

Apply the calculated values to the fraction LCM formula.

\text{LCM} = \frac{45}{2}

Combining the results from the previous steps yields the LCM in fractional form.

5

Convert the resulting fraction into a decimal format.

\frac{45}{2} = 22.5

The question explicitly requires the final answer to be expressed as a decimal.

Key Concept

Calculating the LCM of fractions using the specific formula: LCM of numerators divided by the HCF of denominators.

Question 613Question →

A digital communications system transmits data packets of three specific sizes: $\frac{35}{18}$ MB, $\frac{7}{4}$ MB, and $\frac{49}{24}$ MB. To optimize the network buffer, the engineers must define a standardized base unit size. Every transmitted packet must be an exact multiple of this base unit. To maximize efficiency, what is the largest possible size for this base unit?

\frac{7}{72}

MB

\frac{245}{2}

MB

\frac{7}{2}

MB

\frac{245}{72}

MB

Show answer & explanation

Answer: $\frac{7}{72}$ MB

Answer

The largest possible size for the base unit is

\frac{7}{72}

MB.

To find the largest possible base unit that perfectly divides all three packet sizes, the Highest Common Factor (HCF) of the fractions must be calculated. The formula for the HCF of fractions is HCF(Numerators) / LCM(Denominators). The numerators are 35, 7, and 49, which have an HCF of 7. The denominators are 18, 4, and 24, which have an LCM of 72. Thus, the correct largest base unit is 7/72 MB.

Step-by-Step Solution

1

Identify the required mathematical operation.

Find the Highest Common Factor (HCF) of the three fractions:

\frac{35}{18}

,

\frac{7}{4}

, and

\frac{49}{24}

.

The problem asks for the 'largest possible size' that perfectly divides the sizes of all given packets, which is the definition of HCF.

2

Calculate the HCF of the numerators.

The numerators are 35, 7, and 49. Their HCF is 7.

This value forms the numerator of the final fraction according to the formula.

3

Calculate the Least Common Multiple (LCM) of the denominators.

The denominators are 18, 4, and 24. Their LCM is 72.

This value forms the denominator of the final fraction according to the formula.

4

Combine the results to find the final HCF of the fractions.

\frac{7}{72}

MB.

Applying the formula: HCF of fractions =

\frac{\text{HCF of Numerators}}{\text{LCM of Denominators}}

.

Key Concept

Highest Common Factor (HCF) of Fractions

Question 614Question →

In a family of seven members—A, B, C, D, E, F, and G—the following relationships are known:

- D is the maternal grandfather of E and has exactly three children, all of whom are daughters.
- C is the father of E and has no siblings.
- B is the wife of C.
- F is the mother of G and the sister-in-law of C.

Based on the provided information, which of the following statements must be true?

A is the maternal aunt of E.

E is the grandson of D.

G is the nephew of B.

F is the paternal aunt of E.

Show answer & explanation

Answer: A is the maternal aunt of E.

Answer

The statement that must be true is that A is the maternal aunt of E.

By mapping the family tree, we find that D has three daughters: B (who is E's mother), F (who is G's mother), and A (the remaining family member). Because A is the sister of B, she is the maternal aunt of E.

Step-by-Step Solution

1

Analyze D's relationship to the rest of the family.

D is the maternal grandfather of E. This means E's mother is the daughter of D. We also know D has exactly three children, all daughters.

To establish the top generation and the gender constraints of the second generation.

2

Determine the relationships between B, C, and F.

B is the wife of C, and C is the father of E. Thus, B is E's mother and one of D's three daughters. F is C's sister-in-law. Because C has no siblings, F must be the sister of C's wife (B). Therefore, F is the second daughter of D.

To accurately map the second generation and resolve the 'sister-in-law' connection.

3

Identify the remaining family members.

The family has 7 members. We have identified D (grandfather), B (daughter 1), F (daughter 2), C (B's husband), E (B's child), and G (F's child). The only unassigned member is A. Since D has three daughters, A must be the third daughter.

To complete the family tree using the member count constraint.

4

Evaluate the genders of E and G to verify the statements.

The text states C is the 'father of E' and F is the 'mother of G', but does not provide any pronouns or terms to establish the genders of E and G. Options claiming 'grandson' or 'nephew' cannot be proven true.

To eliminate distractors relying on gender assumptions.

Key Concept

Blood Relations and Multi-generational Family Trees

Estimated Time:2m 0s

Question 615Question →

In a state e-governance initiative, a total of $2,400$ Common Service Centers (CSCs) are distributed across three districts: District P, District Q, and District R. District P contains $30\%$ of the total CSCs. The ratio of the total number of CSCs in District Q to those in District R is $4:3$ . Each CSC is operated by either a male or a female entrepreneur. In District P, $25\%$ of the CSCs are operated by female entrepreneurs. The number of female-operated CSCs in District Q is $50\%$ more than the number of female-operated CSCs in District P. If the total number of female-operated CSCs across all three districts combined is $600$ , what is the ratio of male-operated CSCs in District Q to the female-operated CSCs in District R?

29:11

23:9

23:5

5:23

Show answer & explanation

Answer: $23:5$

Answer

The correct ratio is

23:5

.

Based on the text, District Q has a total of 960 CSCs and District P has 180 female-operated CSCs. Since District Q has 50% more female-operated CSCs than P, it has 270 female-operated CSCs. This leaves 690 male-operated CSCs in Q. Since the total female-operated CSCs across all districts is 600, District R has 600 - (180 + 270) = 150 female-operated CSCs. The ratio of male-operated CSCs in Q to female-operated CSCs in R is 690:150, which simplifies to 23:5.

Step-by-Step Solution

1

Calculate the total number of CSCs in each district.

District P = 720. District Q = 960. District R = 720.

District P has 30% of 2,400 = 720. The remaining 1,680 CSCs are divided between Q and R in a 4:3 ratio. District Q = (4/7) * 1,680 = 960. District R = (3/7) * 1,680 = 720.

2

Calculate the number of female-operated CSCs in Districts P and Q.

Female in P = 180. Female in Q = 270.

Female CSCs in P = 25% of 720 = 180. Female CSCs in Q is 50% more than in P, meaning 180 + (0.50 * 180) = 180 + 90 = 270.

3

Calculate the number of female-operated CSCs in District R.

Female in R = 150.

Total female CSCs = 600. Female in R = Total - (Female P + Female Q) = 600 - (180 + 270) = 600 - 450 = 150.

4

Calculate the number of male-operated CSCs in District Q.

Male in Q = 690.

Total CSCs in Q (960) - Female in Q (270) = 690.

5

Find the requested ratio.

23:5

Ratio of Male in Q to Female in R = 690 : 150. Dividing both sides by 30 gives 23 : 5.

Key Concept

Data Extraction and Percentage Multi-step Calculations in Caselets

Question 616Question →

Six architects—Maya, Noah, Oliver, Piper, Quinn, and Riley—are seated around a triangular table. Three of them sit at the corners of the table facing the center, while the other three sit at the middle of the sides facing outward.

1. Maya sits at a corner.
2. Quinn sits second to the left of Maya.
3. Oliver sits to the immediate right of Noah, but Noah is not seated at a corner.
4. Riley is not an immediate neighbor of Quinn.

Based on the seating arrangement, which of the following statements are true? (Select all that apply)

Select all that apply

Piper sits exactly opposite to Oliver.

Riley sits third to the right of Quinn.

Noah sits to the immediate left of Oliver.

The person sitting to the immediate left of Riley is Maya.

Show answer & explanation

Answer: Piper sits exactly opposite to Oliver.; Riley sits third to the right of Quinn.

Answer

The statements 'Piper sits exactly opposite to Oliver' and 'Riley sits third to the right of Quinn' are true.

Based on the deduced seating arrangement, Piper sits at a middle position opposite to the corner where Oliver sits. Additionally, since Quinn faces the center, counting three positions to the right (counter-clockwise) leads exactly to Riley's position.

Step-by-Step Solution

1

Establish the seating positions and their left/right orientations based on facing directions.

Label positions 1 to 6 clockwise. Corners are 1, 3, and 5 (facing center). Middle positions are 2, 4, and 6 (facing outward). For center-facing (1, 3, 5), left is clockwise and right is counter-clockwise. For outward-facing (2, 4, 6), right is clockwise and left is counter-clockwise.

Tracking relative directions is critical because inward and outward facing seats have reversed left/right orientations.

2

Place Maya and deduce Quinn's position.

Place Maya at corner position 1. Since Maya faces the center, her left is clockwise. Quinn sits second to the left, which is position 3 (also a corner).

Condition 1 and 2 give an absolute starting point and a direct relative position.

3

Determine Noah and Oliver's positions.

Noah is not at a corner, so Noah is at 2, 4, or 6 (facing outward). Oliver is to the immediate right of Noah. Since Noah faces outward, right is clockwise. If Noah were at 2, his right would be 3 (occupied by Quinn). If Noah were at 6, his right would be 1 (occupied by Maya). Thus, Noah must be at 4, placing Oliver at 5 (a corner).

Process of elimination based on Noah's restriction from sitting at a corner and the availability of adjacent seats.

4

Place Riley and Piper in the remaining seats.

The remaining seats are 2 and 6. Riley cannot be an immediate neighbor of Quinn (position 3). Since position 2 is adjacent to 3, Riley must be at position 6. This leaves Piper at position 2.

Condition 4 restricts Riley from sitting next to Quinn, forcing the final placements.

Key Concept

Multi-variable seating arrangement with mixed facing directions (inward and outward) determining relative left/right positioning.

Estimated Time:2m 0s

Question 617Question →

Match each administrative stakeholder conflict scenario in List I with its most appropriate conflict resolution strategy in List II.

Click a left item, then click its matching right item

Show answer & explanation

Answer

Each stakeholder conflict scenario is matched with a resolution strategy that applies proportional administrative intervention, balancing public welfare, institutional coordination, and regulatory compliance.

The correct matches align each specific administrative challenge with a balanced negotiation strategy. Commercial traffic issues require developer accountability and traffic management; inter-agency budget disputes require senior administrative arbitration; vendor congestion demands time-shared spatial regulation; and environmental compliance requires structured, independently audited timelines.

Step-by-Step Solution

1

Analyze the core constraint in each scenario of List I.

Identified traffic impact, inter-departmental budget friction, urban spatial limitations, and regulatory compliance deadlines.

Resolving stakeholder disputes requires pinpointing the underlying operational or administrative root cause.

2

Pair each scenario with the corresponding intervention from List II that offers a balanced solution.

Matched traffic issues to developer parking solutions, agency disputes to administrative arbitration, space limits to time-sharing, and environmental concerns to audited compliance plans.

Effective administration relies on tailored, non-extreme, and sustainable negotiation mechanisms.

Key Concept

Proportional Stakeholder Negotiation and Administrative Conflict Resolution

Question 618Question →

In a survey conducted among $300$ State PSC aspirants, data was collected regarding their enrollment in preparation modules for three optional subjects: Public Administration ( $P$ ), Sociology ( $S$ ), and Geography ( $G$ ). It was found that $140$ aspirants enrolled in Public Administration, $130$ in Sociology, and $120$ in Geography. Further, $50$ aspirants enrolled in both Public Administration and Sociology, $45$ in both Sociology and Geography, and $40$ in both Public Administration and Geography. If $20$ aspirants did not enroll in any of these three subjects, how many aspirants enrolled in EXACTLY ONE optional subject?

195

120

145

165

Show answer & explanation

Answer: 195

Answer

The number of aspirants enrolled in exactly one optional subject is 195.

By applying the inclusion-exclusion principle for three overlapping sets, the number of aspirants enrolled in all three subjects is determined to be

25

. Decomposing each set into disjoint regions gives

75

candidates in Public Administration only,

60

in Sociology only, and

60

in Geography only. Summing these exclusive single-set regions yields

195

.

Step-by-Step Solution

1

Determine the total number of aspirants enrolled in at least one optional subject

Total enrolled

|P \cup S \cup G| = 300 - 20 = 280

.

Candidates not enrolled in any of the three subjects must be excluded from the universe of

300

.

2

Apply the Principle of Inclusion-Exclusion for three sets to find the three-set intersection region

|P \cap S \cap G|

|P \cup S \cup G| = |P| + |S| + |G| - (|P \cap S| + |S \cap G| + |P \cap G|) + |P \cap S \cap G| \implies 280 = 140 + 130 + 120 - (50 + 45 + 40) + |P \cap S \cap G| \implies 280 = 390 - 135 + |P \cap S \cap G| \implies |P \cap S \cap G| = 25

.

To find individual exclusive regions, the central intersection of all three sets must first be determined.

3

Calculate the count of candidates enrolled in exactly two subjects

Only

(P \cap S) = 50 - 25 = 25

; Only

(S \cap G) = 45 - 25 = 20

; Only

(P \cap G) = 40 - 25 = 15

.

Subtracting the triple intersection from pairwise intersections gives the exact counts of elements belonging strictly to two sets.

4

Calculate the count of candidates enrolled in exactly one subject for each subject and sum them up

Only

P = 140 - (25 + 25 + 15) = 75

; Only

S = 130 - (25 + 25 + 20) = 60

; Only

G = 120 - (15 + 25 + 20) = 60

. Total exactly one =

75 + 60 + 60 = 195

.

Subtracting all overlapping regions from each subject set total yields the number of candidates taking only that single subject.

Key Concept

Principle of Inclusion-Exclusion for Three Sets

Question 619Question →

Consider the largest three-digit natural number $N$ which, when successively divided by $6$ , $7$ , and $8$ , leaves remainders of $4$ , $3$ , and $5$ respectively. What is the true positive remainder when the mathematical expression $E = 42^{63} - N$ is divided by $13$ ?

3

7

8

6

Show answer & explanation

Answer: 7

Answer

The correct remainder is 7.

The correct answer is derived by first reconstructing the number through successive division logic to find

N = 904

. Next, applying Fermat's Little Theorem simplifies

42^{63} \pmod{13}

to

1

. The expression becomes

1 - 7 = -6 \pmod{13}

, which correctly maps to a true positive remainder of

7

.

Step-by-Step Solution

1

Determine the algebraic form of the number

N

based on the successive division conditions.

N = 336 \cdot Q_3 + 232

By working backward from the final quotient

Q_3

, we construct the relations:

Q_2 = 8 \cdot Q_3 + 5

,

Q_1 = 7 \cdot Q_2 + 3

, and

N = 6 \cdot Q_1 + 4

. Substituting these yields the general form for

N

.

2

Find the largest three-digit natural number

N

.

N = 904

Substitute increasing integer values for

Q_3

. For

Q_3 = 0

,

N = 232

. For

Q_3 = 1

,

N = 568

. For

Q_3 = 2

,

N = 904

. For

Q_3 = 3

,

N = 1240

(which is four digits). Thus,

904

is the largest valid three-digit number.

3

Calculate the remainder of

42^{63}

divided by

13

.

42^{63} \equiv 1 \pmod{13}

Since

42 = 13 \times 3 + 3

,

42 \equiv 3 \pmod{13}

. By Fermat's Little Theorem,

3^{12} \equiv 1 \pmod{13}

. Breaking down the exponent:

63 = 12 \times 5 + 3

. Thus,

3^{63} \equiv (3^{12})^5 \cdot 3^3 \equiv 1^5 \cdot 27 \equiv 27 \pmod{13}

. Finally,

27 \pmod{13} = 1

.

4

Calculate the remainder of

N

divided by

13

and evaluate the full expression

E

.

E \equiv 7 \pmod{13}

904 = 13 \times 69 + 7

, so

N \equiv 7 \pmod{13}

. Substituting the remainders into the expression:

E \equiv 1 - 7 = -6 \pmod{13}

. To find the true positive remainder, add the divisor:

-6 + 13 = 7

.

Key Concept

Combining Successive Division modeling with Modular Exponentiation and Negative Remainder conversion.

Question 620Question →

If the expression $\frac{3\sqrt{2} - 2\sqrt{3}}{3\sqrt{2} + 2\sqrt{3}}$ can be expressed in the form $a - b\sqrt{6}$ where $a$ and $b$ are rational numbers, what is the exact value of $a + b$ ?

Show answer & explanation

Answer: 7

Answer

The correct value is 7.

The correct answer is derived by multiplying the numerator and denominator by the conjugate

3\sqrt{2} - 2\sqrt{3}

. This rationalizes the denominator to

6

. Expanding the numerator gives

30 - 12\sqrt{6}

. Dividing the numerator by

6

yields

5 - 2\sqrt{6}

. Setting this equal to

a - b\sqrt{6}

identifies

a = 5

and

b = 2

, giving a final sum of

7

.

Step-by-Step Solution

1

Multiply the numerator and denominator by the conjugate of the denominator,

3\sqrt{2} - 2\sqrt{3}

.

\frac{(3\sqrt{2} - 2\sqrt{3})^2}{(3\sqrt{2} + 2\sqrt{3})(3\sqrt{2} - 2\sqrt{3})}

This process, known as rationalizing the denominator, removes the surds from the bottom of the fraction.

2

Expand the numerator using the binomial square formula

(x-y)^2 = x^2 - 2xy + y^2

.

(3\sqrt{2})^2 - 2(3\sqrt{2})(2\sqrt{3}) + (2\sqrt{3})^2 = 18 - 12\sqrt{6} + 12 = 30 - 12\sqrt{6}

Expanding the squared binomial simplifies the top part of the fraction.

3

Expand the denominator using the difference of squares formula

(x+y)(x-y) = x^2 - y^2

.

(3\sqrt{2})^2 - (2\sqrt{3})^2 = 18 - 12 = 6

This guarantees that the denominator becomes a rational number.

4

Divide the terms in the numerator by the denominator.

\frac{30 - 12\sqrt{6}}{6} = 5 - 2\sqrt{6}

Simplifying the fraction allows us to match it to the given form

a - b\sqrt{6}

.

5

Equate the simplified expression to

a - b\sqrt{6}

and solve for

a + b

.

a = 5

,

b = 2

, and

a + b = 7

By direct comparison of rational and irrational parts, we determine the values of

a

and

b

to find their sum.

Key Concept

Rationalizing the denominator using conjugates and expanding binomial expressions involving surds.

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