The greatest common factor of two numbers is the largest number that divides both of them exactly. It is what you divide by to put a fraction in lowest terms in one step instead of several.
Method one, list the factors
GCF of 24 and 36 by listing
factors of 24: 1, 2, 3, 4, 6, 8, 12, 24 factors of 36: 1, 2, 3, 4, 6, 9, 12, 18, 36 common factors: 1, 2, 3, 4, 6, 12 greatest: 12
Reliable, and slow once the numbers grow past two digits.
Method two, prime factorisation
GCF and LCM of 24 and 36 from primes
24 = 2 x 2 x 2 x 3 36 = 2 x 2 x 3 x 3 GCF: take the primes both share 2 x 2 x 3 = 12 LCM: take the highest count of each prime 2 x 2 x 2 x 3 x 3 = 72
One factorisation gives both answers, which is why this method is worth learning even though listing feels easier at first.
A check. The GCF times the LCM always equals the product of the two numbers. Here 12 x 72 = 864 and 24 x 36 = 864.
Using it on fractions
Simplifying 24/36
GCF of 24 and 36 is 12 24 / 12 2 ------- = -- 36 / 12 3
Dividing by the GCF reaches lowest terms in a single step. Dividing by 2, then 2 again, then 3, gets to the same place with three chances to slip.
When the GCF is 1
Numbers whose only common factor is 1 are called relatively prime, or coprime. Nine and sixteen are an example, and neither is prime by itself. A fraction whose numerator and denominator are coprime is already in lowest terms.
Common mistakes
Confusing GCF with LCM
The GCF is never larger than the smaller number. The LCM is never smaller than the larger one. If your GCF came out bigger than both, you found the LCM.
Taking every shared prime once
For the GCF, a prime is taken as many times as it appears in both factorisations. In 24 and 36 the factor 2 appears three times and twice, so it enters the GCF twice.
Check yourself
GCF of 18 and 30.
18 = 2 x 3 x 3 and 30 = 2 x 3 x 5. Shared: 2 x 3 = 6.
Simplify 45/60.
The GCF is 15, giving 3/4.
LCM of 8 and 12.
8 = 2 x 2 x 2, 12 = 2 x 2 x 3. Highest count of each: 2 x 2 x 2 x 3 = 24.
Are 14 and 25 relatively prime?
Yes. 14 = 2 x 7 and 25 = 5 x 5 share no prime, so their GCF is 1.
See also
Sources
- Greatest common factor and least common multiple by prime factorisation, as taught in United States middle grades mathematics. Examples on this page are our own.