Simple Problem 4

Number Theory Level 2

What is the remainder of $2^{10^{9}+7}$ when divided by $10^{9}+7$ ?

2 solutions

Yajat Shamji
Jul 9, 2020

$10^9 + 7 = 1000000007$

$\frac{2^{1000000007}}{1000000007}$

Now using Fermat's little theorem:

$a^{p - 1} = 1 (\text{mod} (p))$

Using $p = a = 2$ :

$2^{2 - 1} = 1 (\text{mod} (2))$

Therefore, the answer is $\fbox 2$

A Steven Kusuman
Jan 23, 2015

Fermat little theorem if(p==prime){ a^(p-1)=1 mod p }