Skip to content

GMAT Remainders, Divisibility Practice Questions

Practice Remainders, Divisibility questions with worked explanations and timing guidance for Quantitative Reasoning.

Five-question preview. Answer 3 now without an account.
Question 1 of 3 free Easy

\(x_1\) and \(x_2\) are each positive integers. When \(x_1\) is divided by 3, the remainder is 1. When \(x_2\) is divided by 12, the remainder is 4. If \(y = 2x_1 + x_2\), which of the following must be true?\n\nI. \(y\) is even.\nII. \(y\) is divisible by 4.\nIII. \(y\) is divisible by 3.

View explanation

Correct answer: D

The statement "when \( x_1 \) is divided by 3, the remainder is 1" can be translated mathematically as follows: There exists an integer n such that \( x_1 = 3n + 1 \).

Similarly, the statement "when \( x_2 \) is divided by 12, the remainder is 4" can be translated mathematically as follows: There exists an integer m such that \( x_2 = 12m + 4 \).

Therefore:
$$\begin{aligned} y &= 2x_1 + x_2 \\ &= 2(3n + 1) + (12m + 4) \\ &= 6n + 2 + 12m + 4 \\ &= 6(n + 2m + 1) \end{aligned}$$

The expression \( (n + 2m + 1) \) must be an integer (since n and m are both integers). Therefore, y is equal to some integer multiplied by 6. This means that y is divisible by 6. Any number that is divisible by 6 must also be divisible by both 2 and 3. Hence, y must be both even and divisible by 3. Consequently, I and III must be true.

The correct answer is D.
Question 2 of 3 free Medium

If (x # y) represents the remainder that results when the positive integer x is divided by the positive integer y, what is the sum of all the possible values of y such that (16 # y) = 1?

View explanation

Correct answer: D

The definition given tells us that when x is divided by y, a remainder of (x # y) results. Since (16 # y) = 1, when 16 is divided by y, a remainder of 1 results.

We can express this algebraically as: 16 = qy + 1, where q is the quotient. Solving for qy gives us: 15 = qy. This means y must be a factor of 15.

The factors of 15 are 1, 3, 5, and 15. However, y cannot equal 1 because the divisor must be greater than the remainder. If y = 1, then dividing 16 by 1 gives a remainder of 0, not 1.

Therefore, the possible values of y are 3, 5, and 15. The sum of these integers is 3 + 5 + 15 = 23.

The correct answer is D.
Question 3 of 3 free Medium

If k and x are positive integers and x is divisible by 6, which of the following CANNOT be the value of \( \sqrt{288kx} \)?

View explanation

Correct answer: B

The value \( \sqrt{288kx} \) can be simplified to \( 12\sqrt{2kx} \). Given that x is divisible by 6, for the purpose of solving this problem x might be restated as 6y, where y may be any positive integer. The expression \( \sqrt{288kx} \) could then be further simplified to \( 12\sqrt{12ky} \) or \( 24\sqrt{3ky} \).

Therefore, each answer choice CAN be a solution if and only if there is an integer y such that \( 24\sqrt{3ky} \) equals that answer choice. The following table shows such an integer value of y for four of the possible answer choices:

$$\begin{array}{|c|c|}\hline y & \text{Solution} \\\hline 1 & 24\sqrt{3k} \\\hline 2 & 24\sqrt{6k} \\\hline 3 & 72\sqrt{k} \\\hline k & 24k\sqrt{3} \\\hline \end{array}$$

The answer choice that cannot be the value of \( \sqrt{288kx} \) is \( 24\sqrt{k} \). For this expression to be a possible solution, y would have to equal \( \frac{1}{3} \), which is not a positive integer. Put another way, this solution would require that \( x = 2 \), which cannot be true because x is divisible by 6.

The correct answer is B.
Question 4 of 5 preview Hard

When the positive integer x is divided by 9, the remainder is 5. What is the remainder when 3x is divided by 9?

Continue Remainders, Divisibility practice

Create a free account to practise five more questions with immediate explanations and saved progress.

Create account and continue Prefer a timed test?
Question 5 of 5 preview Medium

\( 10^{25} \) – 560 is divisible by all of the following EXCEPT:

Continue Remainders, Divisibility practice

Create a free account to practise five more questions with immediate explanations and saved progress.

Create account and continue Prefer a timed test?