Mathematics · Greatest common factor
GCF of 45 and 60
The greatest common factor of 45 and 60 is 15. It is the largest whole number that divides both 45 and 60 with no remainder.
You will also see it written as the highest common factor (HCF) or the greatest common divisor (GCD) of 45 and 60. All three names mean the same thing.
Greatest common factor
15
GCF, GCD or HCF of 45 and 60
Least common multiple
180
LCM of 45 and 60
45/60 simplified
3/4
both divided by 15
Below is the same answer reached three different ways, because different classes teach different methods. Every line is worked out for these two numbers, so you can copy the one your teacher asks for.
Listing the factors of 45 and 60
Factors of 45: 1, 3, 5, 9, 15, 45
Factors of 60: 1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30, 60
Common factors: 1, 3, 5, 15
Greatest of those = 15
The common factors of 45 and 60 are 1, 3, 5 and 15, and that is the complete list. The largest one, 15, is the GCF.
Prime factorization method
45 = 32 × 5 (3 × 3 × 5)
60 = 22 × 3 × 5 (2 × 2 × 3 × 5)
Shared primes at their lowest power: 3 × 5
GCF = 15
Take every prime that appears in both lists, each at the lower of its two powers, and multiply them: 3 × 5 gives 15.
Euclid's algorithm, step by step
60 ÷ 45 = 1 remainder 15
45 ÷ 15 = 3 remainder 0
Last non-zero remainder = 15, so the GCF is 15
Divide the larger number by the smaller, then keep dividing the previous divisor by the remainder. When the remainder reaches zero, the divisor you used last is the GCF. It took 2 steps for 45 and 60.
LCM of 45 and 60, and the fraction 45/60
For any two numbers, the GCF multiplied by the LCM equals the two numbers multiplied together. That gives a quick way to get one from the other, and a way to check both.
GCF × LCM = 15 × 180 = 2,700
45 × 60 = 2,700
They match, so both answers are right
Dividing the top and bottom of 45/60 by 15 reduces it to 3/4, which is as far as it goes. Anything smaller than the GCF would leave the fraction reducible.
Where these two numbers turn up
Forty five and sixty minutes both divide into quarter hour blocks, because 15 is their GCF.
GCF of 45 and 60: FAQs
What is the GCF of 45 and 60?
The GCF of 45 and 60 is 15. It is the largest whole number that divides both of them exactly.
What are the common factors of 45 and 60?
The common factors of 45 and 60 are 1, 3, 5 and 15. The greatest of them is 15.
Is the GCF of 60 and 45 the same as the GCF of 45 and 60?
Yes. The order makes no difference. The greatest common factor of 60 and 45 is 15, exactly the same as the GCF of 45 and 60, because you are looking at the same two numbers either way. The same is true if you see it written as the highest common factor of 60 and 45.
What is the LCM of 45 and 60?
The LCM of 45 and 60 is 180. You can check it with GCF times LCM: 15 times 180 equals 2,700, which is the same as 45 times 60.
Are 45 and 60 coprime?
No. 45 and 60 share 15 as a common factor, so they are not coprime. Coprime numbers have a GCF of exactly 1.
How do you simplify 45/60?
Divide the top and the bottom by their GCF, which is 15. That gives 45 divided by 15 equals 3, and 60 divided by 15 equals 4, so 45/60 simplifies to 3/4.
Which method is quickest for 45 and 60?
Euclid's algorithm, because it reaches the answer in 2 steps and you never have to write out the full factor lists.
Nearby pairs
Any other numbers? The GCF Calculator takes two to ten values and shows the same three methods, and the math calculators page has the rest of the set.
Open the GCF Calculator →
Written & verified by
B.Tech Computer Science · 8 years in web development & SEO · Bhilwara, India
About ·
LinkedIn ·
Report an error