Q: (We denote the least common multiple of a and b by [a,b] or lcm[a,b])

Give a proof by contradiction that if a positive integer n is a common multiple of a and b then [a,b] divides n.

(Hint: Use the division theorem. If [a,b] does not divide n then n=[a,b]q+r where 0<r<[a,b]. Now prove that r is a common multiple of a and b.)

My proof that r is a common multiple of a and b:

r=n-[a,b]q

n=au=av

[a,b]=m=ax=by

r=a(u-xq)=b(v-yq)

Is that correct? Also, how does r being a common multiple of a and b provide a contradiction?