Prove that if gcda b d then gcdak bk dk Solutionlet me ex

Prove that if gcd(a, b) = d, then gcd(a^k , b^k) = d^k .?

Solution

let me explain it with an example.

suppose we have a=2 and b=3

then gcd(2,3)=6

that means d=6

now let us assume k=2

then ak=2*2=4 and bk= 3*2=6

now gcd (ak,bk) = gcd ( 4,6) which is equal to 12

also we can write 12 as 3*6

so dk=3*6 where d=6 and k=3