Elementary Number Theory

Modulo

$$a \equiv b \pmod{n} \iff a - b = qn, \quad \text{where } q \in \mathbb{Z}$$

$$(a \bmod n) + (b \bmod n) \equiv (a + b) \bmod{n}$$

$$(a \bmod n) \cdot (b \bmod n) \equiv (a \cdot b) \bmod{n}$$

Addition and multiplication table

E.g. for integers modulo 4

Common questions

Standard cycle method

Find the last digit of 79999.

Not coprime factors

Find the last digit of 89999.

Operator closure

$$\text{Set } S \text{ is closed under } \triangle \iff \forall x, y \in S, \; x \triangle y \in S$$

Examples

Let S be the set of odd integers and △ be the multiplication. Is S closed under △?

Let S be the set of irrational numbers and △ be the addition. Is S closed under △?