Question

Prove Fermat’s Little theorem. If p is prime and a is any integer, then ${\mathit{a}}^{\mathbf{p}}\mathbf{=}\mathit{a}\left(modp\right)$. [Hint: Let b be the congruence class of a in ${\mathbf{\mathbb{Z}}}_{\mathbf{p}}$and use part (a).

Short answer

It is proved that$a^p=a$(mod p) .

Step-by-step solution

Prove that ap=a(mod p)

Let bbe the congruence class of ${\mathrm{\mathbb{Z}}}_p$.

So, b can be defined as ${\left[a\right]}_p$.

Prove that ap=a(mod p)

If $b\ne 0$. Then,${{\left[a\right]}^p}_p={\left[a\right]}_p{{\left[a\right]}^{p-1}}_p$.

Use the result from part, we can write this as:

Hence, it is proved that $a^p=a\left(modp\right)$.