GCD of Powers

Number Theory Level 1

If $\gcd(30,154)=2$ , find $\gcd(30^5,154^5)$ .

Notation: $\gcd(\, \cdot \, , \cdot)$ denotes the greatest common divisor function.

2 solutions

Madelyn Yu
Jan 1, 2017

$\gcd(30^5,154^5) = gcd(2^5*15^5,2^5*77^5) = 2^5 gcd(15^5,77^5) = 2^5 = 32$

Muhammad Rasel Parvej
Jan 1, 2017

The actual intention of this problem was to help those recognize this following fact who haven't recognized it yet:

For any integer $a$ and $b$ (with not both zero) and any positive integer $n$ , $gcd(a^n,b^n)=(gcd(a,b))^n.$

Given $gcd(30,154)=2$ , applying this fact, $gcd(30^5,154^5)=(gcd(30,154))^5=2^5=32$ .