Chapter 0: 24, 25, 7, 14, 19, 21
Problem 24
- If p is prime and p divides a_1a_2...a_n, prove that p divides a_i for some i
Problem 25
- Use the Generalized Euclid's Lemma to establish the uniqueness portion of the Fundamental Theorem of Arithmetic
Problem 7
- Show that if a and b are positive integers, then ab = lcm(a, b) * gcd (a,b)
Problem 14
- Show that 5n + 3 and 7n + 4 are relatively prime for all n
Problem 19
- Prove that there are infinitely many primes. (hint: use ex. 18)
Problem 21
- For every positive integer n, prove that a set with exactly n elements has exactly 2^n subsets (counting the empty set and the entire set)
