Show that gcda2b2 gcda b2 Show that gcdanbn gcda bn for ea

Show that gcd(a^2,b^2) = gcd(a, b)^2. Show that gcd(a^n,b^n) = gcd(a, b)^n for each n Greaterthanorequalto 1.

Solution

let gcd(a,b) =k

i.e. a and b have the greatest common factor as k

a*a and b*b hence will have k*k =k^2 as the greatest common factor.

So proved that

gcd(a²,b^{2) =} {gcd (a,b)}^2

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b) We can prove by induction for n

Already true for n =2

Let this be true for n

i.e. k^n is the common factor between a^n and b^n

Consider a^(n+1) and b^(n+1)

This is obtained by multiplying a^n by a and b^n by b

Hence one more common factor k is multiplied to the already common factor.

Or k^n*k = k^(n+1) is the common factor for (a^(n+1), b^(n+1))

Proved by induction.