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

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- Dec 5th 2011, 10:42 AMsbacon1991What is the remainder when 11^ 8577314 is divided by 97?
What is the remainder when 11^8577314 is divided by 97?

- Dec 5th 2011, 12:27 PMTheEmptySetRe: What is the remainder when 11^ 8577314 is divided by 97?
Since 97 is prime use Fermat's little theorem. It states that

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

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

So $\displaystyle 8577314 \div 96 = 89347 R 2$

This gives that

$\displaystyle 8577314=96 \cdot 89347 + 2$

So

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

It is all down hill from here