Understanding Why 5 is a Prime Number

Have you ever wondered, ‘Is 5 a prime number?’ Understanding prime and composite numbers is one of the most important basics in maths. The number 5 is considered a prime number because it has exactly two factors: 1 and 5. In this article, we will explain why 5 is a prime number, explore its factors, and learn the difference between prime and composite numbers with simple examples that are easy for students to understand.

Table of Contents

• 5 is a prime number. True or False?
• Proof: Why 5 Is a Prime Number
• Is 5 an Odd Prime?
• Proof 5 is NOT Composite

5 is a prime number. True or False? 

True. To check if 5 is a prime number, let us first check the factors of 5. The factors of 5 are 1 and 5. That is exactly two factors, which meet the definition of a prime number perfectly. A prime number is a natural number greater than 1 that has exactly two distinct positive divisors, 1 and the number itself. Therefore, 5 is prime.

Proof: Why 5 Is a Prime Number

There are a few different ways to confirm that 5 is prime.

Method 1: Factor Method

A number is prime if and only if its only factors are 1 and the number itself. We already confirmed above that 5 has exactly two factors: 1 and 5. Since these are the only two, 5 is prime. QED.

Method 2: Division Method (Trial Division)

To check if a number n is prime, we only need to test divisibility by all primes up to √n. For n = 5, we have √5 ≈ 2.236. The only prime less than or equal to 2.236 is 2. So we just need to check: is 5 divisible by 2?

5 ÷ 2 = 2.5, not exactly divisible. Remainder = 1.

Since 5 is not divisible by any prime up to √5, it has no factors other than 1 and 5.

5 passes the trial division test. Therefore, it is prime.

Method 3: Multiplication Method

We can also verify by trying every possible pair of numbers that could multiply to 5. A number is composite if it can be written as a × b, where both a and b are greater than 1.

There is no way to write 5 = a × b where both a ≥ 2 and b ≥ 2.

Therefore, 5 is prime.

Is 5 an odd prime?

An odd number always has a remainder of 1 when divided by 2, and 5 does exactly that. And since 5 is odd and prime, it is an odd prime.

Proof 5 is NOT composite.

Show that 5 is not a composite number.

For 5 to be composite, it would need to have at least one factor between 2 and 4 (inclusive).

Test each.

  5 ÷ 2 = 2.5,  not a whole number ⇒ 2 is NOT a factor

  5 ÷ 3 = 1.667…,  not a whole number ⇒ 3 is NOT a factor

  5 ÷ 4 = 1.25,  not a whole number ⇒ 4 is NOT a factor

Since no integer between 2 and 4 divides 5 evenly, 5 has no factors other than 1 and 5.

Therefore, 5 is NOT composite. It is prime.