What is the remainder when 7^7 is divided by 19?
Ho do I solve this one?
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remainder problem
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Brent@GMATPrepNow
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Two questions:aarzoo wrote:What is the remainder when 7^7 is divided by 19?
Ho do I solve this one?
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.
Cheers,
Brent
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Matt@VeritasPrep
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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.
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.
