A composite number has at least one prime factor; hence it has at least three factors. If n is a composite number then n is always divisible at least by three distinct numbers.

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## A composite number is divisible at least by one prime

**Question:** Prove that a composite number has at least one prime divisor.

**Solution:**

Let n be a composite number. Thus, by definition, it has a positive divisor other than 1 and n. Let S be the set of these positive divisors of n (which are different from 1 and n).

Clearly, S is non-empty.

By the well-ordering principle of natural numbers, S has a least element, say d.

Note 1< d < n.

**Claim:** d is a prime number.

If d is not a prime, then there is another number d’ that divides d.

So 1< d’ < d < n.

As d’ divides d and d divides n, we have that d’ divides n ⇒ d’ ∈ S.

Thus we arrive at a contradiction to the fact that d is the least element of S. Thus, d is a prime number and the result follows.

Therefore, a composite number has at least one prime divisor.

**Read Also:** If 2^{n}-1 is a prime, then n is a prime

## A composite number is divisible by at least three numbers

**Question:** Prove that a composite number has at least three factors.

**Solution:**

Let n be a composite number.

Then n is divisible by a prime number p by the above theorem.

Also, n is divisible by 1 and n.

As n≠p, the number n is divisible by three numbers 1, p, and n. So we have proved the following:

A composite number has at least three factors. |

**Read Also:** Is 2 a prime number?

## FAQs

**Q1: If n is a composite number, then how many factors n has at least?**

Answer: If n is a composite number, then it has at least three factors.

This article is written by Dr. T. Mandal, Ph.D in Mathematics. On Mathstoon.com you will find Maths from very basic level to advanced level. Thanks for visiting.