A fraction is in simplest form (lowest terms) when its numerator and denominator have no common factor other than 1. Simplifying does not change the value — 84/126 and 2/3 are the same number — it just writes it with the smallest possible whole numbers. This calculator finds the greatest common factor, shows how it was found, and divides it out.
How to use the calculator
- Enter the numerator and denominator. For a mixed number, put the whole part in the left box; only the fraction part is reduced.
- The tape shows the simplest form, the greatest common factor, the decimal value and — for improper fractions — the mixed number.
- The steps show Euclid’s algorithm for the GCF, a prime-factor check, and the division.
The method
Finding the GCF with Euclid’s algorithm
Divide the larger number by the smaller and keep the remainder. Then divide the previous divisor by that remainder, and repeat until the remainder is 0. The last nonzero remainder is the GCF. This works quickly even for very large numbers, which is why the calculator uses it. The Euclid’s algorithm calculator shows the full table of divisions.
Checking with prime factors
Write both numbers as products of primes and multiply the primes they share. This is slower for big numbers but makes it obvious why the answer is right.
Worked example: 84/126
- Euclid: 126 = 1 × 84 + 42, then 84 = 2 × 42 + 0. The last nonzero remainder is 42.
- Prime check: 84 = 2² × 3 × 7 and 126 = 2 × 3² × 7. Shared primes: 2 × 3 × 7 = 42. ✓
- Divide: 84 ÷ 42 = 2 and 126 ÷ 42 = 3, so 84/126 = 2/3.
Quick divisibility tests
Spotting a common factor by eye speeds up mental simplifying:
| Divisible by | Test | Example |
|---|---|---|
| 2 | Last digit is even | 18/24 → 9/12 |
| 3 | Digits add to a multiple of 3 | 51/57 (5+1 = 6, 5+7 = 12) → 17/19 |
| 5 | Last digit is 0 or 5 | 35/40 → 7/8 |
| 9 | Digits add to a multiple of 9 | 27/63 → 3/7 |
| 10 | Both end in 0 | 30/70 → 3/7 |
Special cases
- Zero numerator: 0/5 simplifies to 0, whatever the denominator.
- Negative fractions: the sign moves to the front, so 5/−10 and −5/10 both simplify to −1/2.
- Whole-number results: 24/6 simplifies to 4, because the GCF equals the denominator.
- Already simplified: 17/1 or 8/15 have a GCF of 1 and stay as they are.
Simplifying is the last step of almost every fraction calculation — adding, multiplying or converting — so the same GCF method appears in the fraction calculator. To list fractions equal to a given one instead of reducing it, use the equivalent fractions calculator, and to see a number’s prime factors on their own, try the prime factorization calculator.
Frequently asked questions
How do you simplify a fraction?
Find the greatest common factor (GCF) of the numerator and denominator, then divide both by it. For 84/126 the GCF is 42, so 84/126 = (84 ÷ 42)/(126 ÷ 42) = 2/3.
How do I know a fraction is in lowest terms?
A fraction is fully simplified when the only whole number that divides both the numerator and the denominator is 1. For example 8/15 is in lowest terms because 8 = 2³ and 15 = 3 × 5 share no prime factor.
Can I simplify in several smaller steps?
Yes. Dividing 84/126 by 2 gives 42/63, then by 3 gives 14/21, then by 7 gives 2/3. Dividing by the GCF does it in one step, but any sequence of common factors reaches the same answer.
How do I simplify a mixed number?
Leave the whole number alone and reduce the fraction part: 3 6/8 becomes 3 3/4. If the fraction part is improper, such as 2 10/4, carry the extra wholes: 10/4 = 2 2/4, so the answer is 4 1/2.