(15 points) Problem 3.1.4
The least common multiple of two naturals x and y is the smallest natural that both x and y divide. For example, lcm(8, 12) = 24 because 8 and 12 each divide 24, and there is no smaller natural that both 8 and 12 divide.
(a) Find the least common multiple of 60 and 339.
(b) Find the least common multiple of 233254 and 223453.
(c) Describe a general method to find the least common multiple of two naturals, given their factorization into primes (and assuming that the factorization exists and is unique).
(15 points) Problem 3.3.4
We have defined the factorial n! of a natural n to be the product of all the naturals from 1 through n, with 0! being defined as 1. Let p be an odd prime number. Prove that (p − 1)! is congruent to −1 modulo p. (Hint: Pair as many numbers as you can with their multiplicative inverses.)
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