Let a b c element of Z Prove that if gcda b 1 and at and bt

Let a, b, c element of Z. Prove that if gcd(a, b) = 1 and a|t and b|t, then ab|t

Solution

a|t

Hence, t=ma

b|t hence, t=nb

But, gcd(a,b)=1

ie a and b have no prime factors in common

ma=nb

HEnce, b divides ma

But b and a have no factors in common

Hence, b divides m

So, m=kb

So, t=ma=kba=kab

HEnce, ab|t