Why 2 is the only Even Prime Number

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Even Prime Proof

1. What is a Prime Number ?

A prime number has exactly two distinct positive factors: $$1\text{ and itself}$$

Ex: 5 → Factors: 1, 5 → Exactly 2 factors → Prime

2. What is an Even Number ?

An even number is any number that is divisible by 2.

Ex: $$2,4,6,8,10,12,\dots$$

3. Check the Number 2

Factors of 2: $$1,2$$

It has exactly two factors.

Therefore: 2 is prime​. And 2 is also even.

4. What About Any Even Number Greater Than 2 ?

Let $n$ be an even number where: $$n>2$$

Since $n$ is even, it is divisible by 2.

Therefore, its factors include: $$1,2,n$$

That's already at least three factors.

So, $n$ cannot be prime.

5. Examples : 

Note: Prime: exactly 2 factors → 1 and itself
Composite: more than 2 factors

Factors: $$1,2,4$$→ 3 factors → Not prime

Factors: $$1,2,3,6$$→ 4 factors → Not prime

Factors: $$1,2,4,8$$→ 4 factors → Not prime

Trick: Every even number greater than 2 is composite.​

Therefore: 2 is the ONLY even prime number.​

Note: 2 = Even + Prime. 4, 6, 8, 10, ... = Even + Composite