If a, b, and c are consecutive integers, wher… | GMAT Sprint

Free Practice Question

PS · Problem Solving

705-805

Divisibility/Multiples/Factors

If a, b, and c are consecutive integers, where 10 < a < b < c, which of the following could be the remainder when $a^c$ is divided by b?

Answer Choices

Correct answer marked below

I only

II only

III only

I and II only

Correct

I and III only

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Explanation

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Official Explanation

Official

We have three consecutive integers a, b, and c with 10 < a < b < c. So we can write a = b - 1 and c = b + 1.

We want the remainder when

(b - 1)^{(b + 1)}

is divided by b.

When you expand this expression, all terms except the last one will be multiples of b and thus will be divisible by b. The last one will be

(-1)^{(b + 1)}

.

Hence, the question essentially boils down to finding the remainder when

(-1)^{(b + 1)}

is divided by b.

If b is odd, then b + 1 is even, so

(-1)^{(b + 1)} = 1

. And 1 divided by b will give the remainder of 1.

If b is even, then b + 1 is odd, so

(-1)^{(b + 1)} = -1

. And -1 divided by b will give the remainder of b - 1. For example, -1 divided by 20 gives the remainder of 20 - 1 = 19.

Answer: D.

P.S. The process for finding the remainder when dividing a negative integer by a positive integer follows the same principles as when dividing a positive integer by a positive integer.…

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What this question tests

What it tests

Your understanding of divisibility rules, least common multiples, and greatest common factors.

Common trap

Forgetting that 1 is not prime and that every integer divides 0.

If a, b, and c are consecutive integers, where 10 < a < b < c, which of the following could be the remainder when aca^cac is divided by b?