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remainder problem

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Source: — Problem Solving |

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by Brent@GMATPrepNow » Sun Sep 25, 2016 7:02 am
aarzoo wrote:What is the remainder when 7^7 is divided by 19?
Ho do I solve this one?
Two questions:
1) Where are the 5 answer choices that always accompany GMAT questions?
2) What is the source of this question?

Unless I'm missing something, this question really isn't a GMAT-quality question, since it requires too much number crunching (e.g., looking for a pattern by finding the remainder when 7^2, 7^3 and 7^4 are divided by 19)

We could use some modular arithmetic (https://en.wikipedia.org/wiki/Modular_arithmetic) to solve this, but I believe this would be out of scope for the GMAT.


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Brent
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by Matt@VeritasPrep » Thu Sep 29, 2016 7:26 pm
I'd use some smaller powers to help you.

7� = 7³ * 7³ * 7

Now look for powers of 19 close to these. 7³ = 343, and 19² = 361, so those are good candidates. Notice that 361 - 19 = 342, so 343 / 342 has remainder 1. Thus 7³ has remainder 1 by 19, and

7� = 7³ * 7³ * 7 = (remainder 1) * (remainder 1) * (remainder 7) = remainder 7.
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