Recognising an irreducible fraction

Definition

A fraction a/b is said to be irreducible when it cannot be simplified any further, that is, when GCD(a, b) = 1. We then say that a and b are relatively prime.

How do you check whether a fraction is irreducible?

Simply calculate the GCD of the numerator and the denominator:

• If GCD = 1, the fraction is already irreducible.
• If GCD > 1, the fraction can still be simplified.

Example: Is 7/15 irreducible?

• Divisors of 7: 1, 7
• Divisors of 15: 1, 3, 5, 15
• Only common divisor: 1, so GCD(7, 15) = 1 -> 7/15 is irreducible.

Another example: Is 9/12 irreducible?

• GCD(9, 12) = 3, so no, it can still be simplified: 9/12 = 3/4.

A quick tip

If both the numerator and the denominator are even, the fraction is never irreducible (it can at least be divided by 2). Similarly, if both end in 0 or 5, it can be divided by 5.

Common pitfalls to avoid

• Assuming a fraction is irreducible simply because the numerator is a prime number: you must check that it does not also divide the denominator (for example, 7/14 is not irreducible, as GCD(7, 14) = 7).
• Forgetting to check the simplified fraction: sometimes you need to simplify several times in a row if you haven’t used the GCD straight away.
• Confusing an ‘irreducible fraction’ with a ‘fraction less than 1’: irreducibility refers only to simplification, not to the size of the fraction (5/3 may well be irreducible).