What "Lowest Terms" Actually Means

Two integers are coprime - or relatively prime - when the only positive integer that divides both of them is 1. A fraction is in lowest terms when its numerator and denominator are coprime. Simple as that.

8/12 is NOT in lowest terms - both are divisible by 4
2/3 IS in lowest terms - GCF(2, 3) = 1

15/25 is NOT in lowest terms - both are divisible by 5
3/5 IS in lowest terms - GCF(3, 5) = 1

Notice that 8/12 and 2/3 are equivalent fractions - the same spot on the number line. Simplifying doesn't change what the fraction equals, it just expresses it more cleanly. In most math contexts an unsimplified answer isn't accepted as a final answer, so this step genuinely matters.

Finding the GCF of Numerator and Denominator

The Greatest Common Factor (GCF) of two numbers is the largest integer that divides both without leaving a remainder. There are two solid ways to find it:

Method 1: List All Factors

Write out every factor of each number, then find the largest one they share.

GCF of 42 and 56:
Factors of 42: 1, 2, 3, 6, 7, 14, 21, 42
Factors of 56: 1, 2, 4, 7, 8, 14, 28, 56
Common factors: 1, 2, 7, 14
GCF = 14

Method 2: Prime Factorisation

Break each number down to its prime factors. The GCF is the product of all the primes they have in common.

GCF of 60 and 84:
60 = 2 × 2 × 3 × 5
84 = 2 × 2 × 3 × 7
Shared primes: 2 × 2 × 3 = 12

Prime factorisation works better with large numbers. Listing every factor of 360 is tedious. Breaking it into primes is manageable.

Dividing Both Parts by the GCF

Once you have the GCF, divide the numerator by it and the denominator by it. Both parts. Dividing only one side would change the value of the fraction completely.

Simplify 42/56:
GCF(42, 56) = 14
42 ÷ 14 = 3
56 ÷ 14 = 4
Result: 3/4
Simplify 60/84:
GCF(60, 84) = 12
60 ÷ 12 = 5
84 ÷ 12 = 7
Result: 5/7
Simplify 120/180:
GCF(120, 180) = 60
120 ÷ 60 = 2
180 ÷ 60 = 3
Result: 2/3

If you can't spot the GCF right away, divide by any common factor you can see, then check again. You'll get to the same answer in more steps, but you'll get there.

Checking If a Fraction Is Already Simplified

Before spending time hunting for a GCF, a quick check can tell you whether simplification is even needed:

  • If one number is prime and the other is not a multiple of it, GCF = 1 - already simplified.
  • If both numbers are odd, 2 is not a common factor - look only at odd primes (3, 5, 7, ...).
  • If the numerator is 1, the fraction is always in lowest terms regardless of the denominator.
Is 13/39 in lowest terms?
13 is prime. Is 39 divisible by 13? 39 ÷ 13 = 3 → Yes.
GCF = 13 → NOT simplified. Result: 1/3

Is 17/51 in lowest terms?
17 is prime. 51 ÷ 17 = 3 → GCF = 17 → NOT simplified. Result: 1/3

Is 11/17 in lowest terms?
Both are prime. GCF(11, 17) = 1 → Already in lowest terms

Simplifying Improper Fractions

An improper fraction has a numerator larger than its denominator, like 28/12. You simplify it exactly the same way - find the GCF and divide both parts - then optionally convert to a mixed number if that's what the problem calls for.

Simplify 28/12:
GCF(28, 12) = 4
28 ÷ 4 = 7
12 ÷ 4 = 3
Simplified improper fraction: 7/3
As a mixed number: 2 and 1/3
Simplify 135/90:
GCF(135, 90) = 45
135 ÷ 45 = 3
90 ÷ 45 = 2
Simplified: 3/2
As a mixed number: 1 and 1/2
Simplify 252/168:
168 = 2³ × 3 × 7
252 = 2² × 3² × 7
GCF = 2² × 3 × 7 = 84
252 ÷ 84 = 3
168 ÷ 84 = 2
Simplified: 3/2

Proper or improper, the process is the same: find the GCF of numerator and denominator, then divide both. The CalcSolver Pro GCF calculator finds the greatest common factor instantly so you can skip straight to the division step.