Question

find the remainder when 2^1000 is divided by 17 using Fermat's theorm​

Answer 1

Given:

$\frac{2^{1000}}{17}$

To find:

The remainder by the Fermat's Theorem.

Solution:

Fermat's theorem in mathematics is defined as the For any a and p coprime numbers the remainder of the given expression always be 1.

$\frac{a^{(p-1)}}{p}$

We have given the expression as:

• $\frac{2^{1000}}{17}$

Reducing the power of the 2 for applying the Fermat's theorem we get

• $\frac{(2^{62})^{16}.2^{8}}{17}$

By the Fermat's Theorem

• $\frac{(2^{62})^{16}}{17} = 1$

we get:

• $\frac{2^{8}}{17}$
• $\frac{256}{17}$

Here we will get the remainder as 1 only.

So,

The remainder for $\frac{2^{1000}}{17}$ is 1