If a|k and b|k prove that lcm(a,b)|k.

If 'a' divides 'k' and 'b' divides 'k' prove that the least common multiple of (a,b) divides k.

Let the greatest common divisor of a and b be g. Then a=cg and b=dg for some integers c and d. Then if a divides k and b divides k we must have k=ncdg for some integer n. But the least common multiple of a and b is cdg, so the least common multiple must divide k.