Question

Find the remainder when ${17}^{36}$ is divided by 36.
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Answer 1

Approach 1 -

Go by frequency method.

$\frac{17}{36}$ gives a remainder $=17$

$\frac{{17}^2}{36}$ gives a remainder $=1$

Frequency $=2$

Odd powers remainder $=17$, even powers $=1$

Approach 2 -

Euler’s number of $36$ is $36\times \frac{1}{2}\times \frac{2}{3}=12$

$36=12\times 3.$ Hence, from Fermat's theorem $\frac{{17}^{36}}{36}{|}_R=1$