1. Find the gcd of a = 163438 and b = 16150 and express the gcd as a linear combination of a and b.

2. How many zeros does 321! end in (when written in decimal notation)?

3. (a) Give an example of two different positive integers with (exactly) 600 divisors.

We know that pm, where p is a prime and m is a positive integer has exactly m + 1 divisors. So, 2599 and 3599 are two different positive integers with exactly 600 divisors.

(b) What is the smallest positive integer with (exactly) 600 divisors that you can find? (You do not have to find the smallest integer with 600 divisors, just try to find one as small as you can – the smaller the better!)

4. For what integers n is n2 ? 1 a prime number? State and prove a theorem that answers this question.

5. Prove that the product of two consecutive integers, both greater than 2, has at least three (not necessarily distinct) prime factors.

ln 2

6. Prove that ln 3 is irrational.