1. What is the smallest positive integer that has exactly k divisors, for 1 k 6? 2. Prove that gcd(m, n) lcm(m, n) = m n, and use this identity to express lcm(m, n) in terms of lcm(n mod m, m), when...

1. What is the smallest positive integer that has exactly k divisors, for 1

k

6?

2. Prove that gcd(m, n) lcm(m, n) = m n, and use this identity to express lcm(m, n) in terms of lcm(n mod m, m), when n mod m ≠ 0.

3. Let π(x) be the number of primes not exceeding x. Prove or disprove: