Find the greatest common divisor of the numbers 2^(120) − 1 and 2^(100) − 1

Question

Find the greatest common divisor of the numbers 2¹²⁰ − 1 and 2¹⁰⁰ − 1

Answer 1

It is a theorem that gcd(2^x-1,2^y-1)=2^gcd(x,y)-1 (it is usually proven using the Euclidean algorithm)

This means that for your particular question, we can first compute gcd(120,100)=20 and then conclude that

gcd(2¹²⁰-1,2¹⁰⁰-1)=2²⁰-1