. Recall that if n and m are nonnegative integers then the least common multiple of n and m, Icm(n., m), is (usually the smallest positive integer that is divisible by both n and m. There is a special case: For every integer k e N, Icm(k,0) = Icm(0, k) = kso that Icm(0,0) = 0. If & > 1 then Icm(k, 1) = Icm(1,k) = k as well. The greatest common divisor of n and m, ged(ni, m), is the largest integer that divides both n and m. There is an exception: ged (0,0) will be defined to be 0. If k > 1 then ged(k,0) = ged (0, k) = k - and this follows by the above definition of god (nm). If k > 1 then godk, 1) = ged(1,k) = 1. 1 A bit more information will be helpful when n,m > 2. Recall that every positive integer that is greater than or equal to two has a factorization as a product of primes - so that if n,m > 2 then there exists an integer k > 1, a set of distinct primes P1, P2, ... Ple and unique nonnegative integers 1,C2, ..., et and f1, f2, ..., fa such that n=pil pe X... XP and m=pf' x pub2 x ... Xola (a) What are the prime factorizations of Icmn, m) and god (no, m)? Hint They both involve the above set of primes and integer exponents such that the coefficient for p is, somehow, related to e, and fi. (b) How large can icm(n,m) be? Hint There is an upper bound that is a simple function of n and m