Introduction to Number Theory

Suppose $a$, $b$, and $c$ are integers. Prove that if $a \mid b$, then $a \mid bc$.

Write out the remainder classes for $n = 4$.

Let $a$, $b$, $c$, and $n$ be integers. Prove that if $a \equiv b \pmod{n}$ and $c \equiv d \pmod{n}$, then $a-c \equiv b-d \pmod{n}$.

Find the remainder of $3^{456}$ when divided by

1. 2.

2. 5.

3. 7.

4. 9.

Determine which of the following congruences have solutions, and find any solutions (between 0 and the modulus) by trial and error.

1. $4x \equiv 5 \pmod 6$.
2. $6x \equiv 3 \pmod 9$.
3. $x^2 \equiv 2 \pmod 4$.

Solve the following congruence $5x + 8 \equiv 11 \pmod{22}$. That is, describe the general solution.

Solve the congruence: $4x \equiv 24 \pmod{30}$.

I'm thinking of a number. If you multiply my number by 7, add 5, and divide the result by 11, you will be left with a remainder of 2. What remainder would you get if you divided my original number by 11?

Solve the following linear Diophantine equation, using modular arithmetic (describe the general solutions). .