GCF and LCM Calculator
The greatest common factor is what you divide by to simplify a fraction. The lowest common multiple is the denominator you need to add two. They are the two halves of the same idea.
Greatest common factor
12
- Greatest common factor
- 12
- Lowest common multiple
- 240
- Factors of 48
- 1, 2, 3, 4, 6, 8, 12, 16, 24, 48
- Factors of 60
- 1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30, 60
- Shared factors
- 1, 2, 3, 4, 6, 12
- The greatest common factor is what you divide by to put a fraction in lowest terms. The lowest common multiple is the denominator you need to add two fractions.
- Multiplying the two together gives the product of the original numbers — GCF times LCM always equals a times b, which is a quick way to check either one.
About factors and multiples
A factor divides into a number exactly. A multiple is what you get by multiplying it. The greatest common factor of 48 and 60 is 12 — the largest number that goes into both — and their lowest common multiple is 240, the smallest number both go into.
The two are connected by a rule worth knowing: the GCF times the LCM always equals the two numbers multiplied together. 12 × 240 is 2,880, and so is 48 × 60. That makes each a quick check on the other.
Finding the GCF by listing factors works and is slow. Euclid's algorithm is faster and two and a half thousand years old: divide the larger by the smaller, keep the remainder, repeat with the smaller and the remainder until the remainder is zero. The last non-zero value is the answer, and it takes a handful of steps however large the numbers are.
Where these actually come up is fractions. Reducing 48/60 means dividing both by their GCF of 12, giving 4/5. Adding fractions with denominators of 48 and 60 means rewriting both over their LCM of 240. Two numbers whose GCF is 1 are coprime, and their LCM is simply their product.
What it works out
- Greatest common factor and lowest common multiple
- Every factor of both numbers
- The factors they share
The formula
GCF by Euclid: replace (a, b) with (b, a mod b) until b is zero LCM = a × b ÷ GCF
Euclid on 48 and 60: 60 divided by 48 leaves 12. Then 48 divided by 12 leaves 0. The last non-zero remainder was 12, so that is the greatest common factor.
It works because any number dividing both 48 and 60 must also divide their difference, so the problem shrinks at every step without losing the answer. Four steps would handle numbers in the millions.
The lowest common multiple then follows without any further searching: 48 × 60 ÷ 12 is 240.
Listing the factors is slower but shows what is happening. 48 has ten factors and 60 has twelve; six of them are shared, and 12 is the largest of those. The shared list is always the full factor list of the GCF itself, which is a neat consequence rather than a coincidence.
- Factor
- A number that divides in exactly, leaving no remainder.
- Multiple
- A number this one divides into exactly.
- Coprime
- Two numbers whose only common factor is 1. Their LCM is their product.
A worked example
The numbers 48 and 60.
That works out to 12 .
- Greatest common factor
- 12
- Lowest common multiple
- 240
- Factors of 48
- 1, 2, 3, 4, 6, 8, 12, 16, 24, 48
- Factors of 60
- 1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30, 60
- Shared factors
- 1, 2, 3, 4, 6, 12
Questions
What is the greatest common factor?
The largest number that divides into both without a remainder. For 48 and 60 it is 12. It is what you divide a fraction by to put it in lowest terms.
What is the lowest common multiple?
The smallest number both divide into. For 48 and 60 it is 240. It is the denominator you need to add two fractions together.
How are the two related?
GCF times LCM always equals the two numbers multiplied together. 12 × 240 is 2,880, and so is 48 × 60. Each is a quick check on the other.
What is Euclid's algorithm?
A fast way to find the GCF: divide the larger by the smaller, keep the remainder, and repeat until the remainder is zero. It takes a handful of steps however large the numbers, and it is about 2,300 years old.
What does coprime mean?
Two numbers whose only common factor is 1 — 17 and 5, for instance. They share nothing to cancel, so their lowest common multiple is just their product.
Why do I need these for fractions?
The GCF simplifies one fraction; the LCM lets you add two. Reducing 48/60 divides both by 12 to give 4/5, and adding thirds to quarters means rewriting both over twelve.