What is the remainder when 11^ 8577314 is divided by 97?

**sbacon1991** · Dec 5th 2011, 10:42 AM

What is the remainder when 11^8577314 is divided by 97?

**TheEmptySet** · Dec 5th 2011, 12:27 PM

Originally Posted by sbacon1991

What is the remainder when 11^8577314 is divided by 97?

Since 97 is prime use Fermat's little theorem. It states that

$a^{p-1} \equiv 1 \text{mod}p$ so this gives that

$11^{96} \equiv 1 \text{mod}97$

So $8577314 \div 96 = 89347 R 2$

This gives that

$8577314=96 \cdot 89347 + 2$

So

$11^{8577314} \equiv 11^{96 \cdot 89347 + 2} \equiv 11^2 \text{mod}97$

It is all down hill from here

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What is the remainder when 11^ 8577314 is divided by 97?

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