Does Every Integer Greater Than 1 Have a Prime Divisor?

Yes, every integer greater than 1 has a prime divisor.

Here's why:

Any integer n greater than 1 is either prime itself or composite.

• If n is prime: Then n is divisible by itself, and since n is prime, it is a prime divisor of itself.

• If n is composite: Then n can be factored into two smaller integers, a and b, such that n = a b, where 1 < a < n and 1 < b < n. If a is prime, then we have found a prime divisor of n. If a is not prime, then it can be further factored into smaller integers. We can continue this process of factoring. Because each factor is smaller than the previous one, this process must eventually terminate in a prime factor. This prime factor will be a divisor of a, and since a is a divisor of n, the prime factor will also be a divisor of n.

Example:

Let's take the integer 12.

1. 12 is composite, so we can factor it as 12 = 3 * 4.
2. 3 is prime, so it is a prime divisor of 12. We could stop here.
3. 4 is composite, so we can factor it as 4 = 2 * 2.
4. 2 is prime, so it is a prime divisor of 4 (and therefore also a prime divisor of 12).

Therefore, every integer greater than 1 has at least one prime divisor.