13. The least common multiple of two non-zero integers a and b is the unique positive inte- ger m such that (i) m is a common multiple, i.e. a divides m and b divides m, (ii) m is less than any other common multiple: a divides n and b divides n m<n. We denote the least common multiple of a and b by [a, b] or 1cm[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. [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.} This means that ab/[a,b] is an integer. Prove that this integer is a common divisor of a and b. Deduce that ab/[a, b] < (a, b), the greatest common divisor of a and b. Prove that ab/(a, b) is a common multiple of a and b. Deduce that (a, b) [a, b] = ab if a and b are positive.