Question

a) Show that [a,b] | ab.

Defn: Let m e Z. We say that m is a common multiple of a and b if alm and bm. Defn: We say that l E Z is the least common multiple of a and b if it's a common multiple of a and b and 0 < t < m for all common multiples, m, of a and b. We denote the least common multiple by [a, b], or lcm[a, b]. Notice: Since O is a common multiple of any pair of integers, we actually have to specify that the least common multiple should not be 0.

b) Let d be a common divisor of a and b. Show that $\left [ a,b \right ]|\frac{ab}{d}$ .

c) Prove that (a,b)*[a,b] = ab.

d) Prove that if c is a common multiple of a and b, then $\exists k \epsilon \mathbb{Z}$ such that k[a,b] = c.

e) Suppose that c is a common multiple of a and b. Show that ab | (a,b)*c

Answer 1