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## Homework Statement

Evaluate (16^1000 - 18^2000)(mod 17)

## Homework Equations

I'm not sure how to go about doing this, but I realise it has something to do with the pattern from the last digit obtained from the 2 large numbers

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- Thread starter valianth1
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- #1

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Evaluate (16^1000 - 18^2000)(mod 17)

I'm not sure how to go about doing this, but I realise it has something to do with the pattern from the last digit obtained from the 2 large numbers

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Dick

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## Homework Statement

Evaluate (16^1000 - 18^2000)(mod 17)

## Homework Equations

I'm not sure how to go about doing this, but I realise it has something to do with the pattern from the last digit obtained from the 2 large numbers

## The Attempt at a Solution

That's really not a great realization. I would think about what 16 and 18 are mod 17.

- #3

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16(mod 17) = -1 and 18(mod 17) = 1, how do I go from here?

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Dick

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16(mod 17) = -1 and 18(mod 17) = 1, how do I go from here?

What's (-1)^1000-1^2000? (x mod 17)^n mod 17=(x^n) mod 17. Right?

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- #6

I like Serena

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Consider that (-1)

Btw, since 17 is prime you can also apply Fermat's little theorem:

For any prime p: a

Rewrite 16

- #7

Dick

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16 and (-1) are congruent mod 17 since they differ by 17. So (-1)^1000=16^1000 mod 17. You can use either form in your calculation. Which one makes it easy?

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