For a positive integer n, what is the remaind… | GMAT Sprint

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PS · Problem Solving

655-705

Exponents/Powers

For a positive integer n, what is the remainder when $n^7 - n$ is divided by 42?

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Official Explanation

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To find the remainder of

n^7 - n

when divided by 42, we can factor it as

n(n^6 - 1)

. The expression

n^6 - 1

can be factored further using the difference of squares:

(n^3 - 1)(n^3 + 1)

. The factors

n(n - 1)(n + 1)(n^2 + 1)(n^2 - n + 1)

show that for any positive integer n, at least one of the factors will be divisible by 2…

Comments

JasonMar 14, 2025

Thankss

SiddharthNov 12, 2024

thanks for the answer

ChiragJan 19, 2024

Greatt

What this question tests

Your grasp of exponent rules — multiplying and dividing powers, power-of-a-power, and negative/zero exponents.

Adding exponents when multiplying bases that have the same exponent, or confusing (x^a)^b with x^(a·b).