Practice Midterm Problems

1. Find all solutions to the linear congruences: a)

$$4x + 22 \equiv 0 (mod 12)$$

b)

$$3x + 15 \equiv 0 (mod 33)$$

2. Determine all solutions to the simultaneous system

$$4x + 11 \equiv 3 (mod 18)$$

$$6x + 9 \equiv 33 (mod 45)$$

3. Solve

$$x^{22} - x^2 + 2x +2 \equiv 0 (mod 11)$$

4. Solve

$$2x^2 - x + 14 \equiv 0 (mod 125)$$

5. Schumer Q. 7, p. 58
Prove that for any fixed $m \geq 2$, the equation $\tau(n)=m$ has infinitely many solutions.

6. Schumer Q. 3, p. 62
Show that if f is a multiplicative function then f(1)=1.