MIDTERM EXAM

Elementary Number Theory – Math 392

Instructor: Eyal Goren. Date: October 21, 1998. Time: 1 hour, 30 minutes.

Answer the following questions. Notebooks, books and calculators are not allowed. Do NOT use a pencil.

1. Prove that there are infinitely many primes.

2. Prove that ^{3}\sqrt{2} is irrational.

3. Let d be the greatest common divisor of 33 and 93. Find d. Find x, y such that 33x + 93y = d.

4. Let

α(n) = ∑_{d|n} μ (\frac{n}{d}) μ(d)^{2}.

Find α(10^{a}) for every a ∈ ℕ. (Hint: “Think multiplicative”).

5. Let ϕ be Euler’s function. Prove that ϕ(n) = n − 1 if and only if n is a prime number.

GOOD LUCK