Question to Prove a Theorem

Topic : Theorem

Theorem : Prove that if gcd(a,p²)=p and gcd(b,p²)=p² then gcd(ab,p^4)=p³. where a and b are integers and p is a prime number.

Solution :
GCD (a, p²) = p
implies that a ia a multiple of p or p is a divisor of a.
So let a = kp
where k is a constant
Similarly GCD(b, p²)=p²
implies that b is the multiple of p² or p² is a divisor of b
So let b = mp²
where m is a constant
So a = kp and b = mp²
ab = kp . mp²
ab = kmp³
implies that ab is a multiple of p³ and km is the constant
So greatest common dividor of kmp³ and p^4 is p³
Hence GCD (ab, p^4) = p³
Hence proved.