GCF for 36, 60 and 84

"Greatest Common Factor of 3 Numbers" Calculator
Greatest Common Factor of
,
and

What's the Greatest Common Factor (GCF) of 36, 60 and 84?

Answer

GCF of 36, 60 and 84 is
12

(Twelve)

12 is the largest positive integer that divides 36, 60 and 84 without a remainder — every other common divisor of the three divides it too.

Finding the GCF of 36, 60 and 84 by Prime Factorization

Split all three numbers into prime factors, then keep every prime that appears in all three of them, each taken as many times as it appears in the poorest factorization. Multiplying what is left gives the greatest common factor:

All Prime Factors of 36: 2 × 2 × 3 × 3 (exponent form: 22 × 32)

All Prime Factors of 60: 2 × 2 × 3 × 5 (exponent form: 22 × 3 × 5)

All Prime Factors of 84: 2 × 2 × 3 × 7 (exponent form: 22 × 3 × 7)

Prime factor366084Lowest power
222222222
332333
5—5——
7——7—

GCF = 2 × 2 × 3 = 12

Checking the GCF of 36, 60 and 84 Two Numbers at a Time

The greatest common factor is associative, so three numbers can be handled as two pairs — and this is what makes the Euclidean algorithm usable here: run it on the first pair, then run it again on that result and the third number.

GCF(a, b, c) = GCF(GCF(a, b), c)

GCF(36, 60) = 12

GCF(12, 84) = 12

GCF(36, 60, 84) = 12

The answer checks out by division — all three quotients come out whole:

36 ÷ 12 = 3

60 ÷ 12 = 5

84 ÷ 12 = 7

3, 5 and 7 have no common factor left — that is what makes 12 the greatest one.

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See Also

About "Greatest Common Factor of 3 Numbers" Calculator

This calculator finds the Greatest Common Factor (GCF) of three numbers at once, without doing it in two passes. For example, it can help you find out what's the Greatest Common Factor (GCF) of 36, 60 and 84? (The answer is: 12). Enter the first number (e.g. '36'), the second (e.g. '60') and the third (e.g. '84'), then hit the 'Calculate' button.
The Greatest Common Factor of three numbers — also called the greatest common divisor (GCD) or highest common factor (HCF) — is the largest positive integer that divides all three of them with zero remainder
"Greatest Common Factor of 3 Numbers" Calculator
Greatest Common Factor of
,
and

GCF Table

Number 1Number 2Number 3GCF
3660693
3660702
3660711
36607212
3660731
3660742
3660753
3660764
3660771
3660786
3660791
3660804
3660813
3660822
3660831
36608412
3660851
3660862
3660873
3660884
3660891
3660906
3660911
3660924
3660933
3660942
3660951
36609612
3660971
3660982

FAQ

What's the Greatest Common Factor (GCF) of 36, 60 and 84?

GCF of 36, 60 and 84 is 12

What prime factors do 36, 60 and 84 share?

All three factorizations contain 2 × 2 × 3, and multiplying those shared primes gives 12 — exactly the greatest common factor.

Can you get the GCF of 36, 60 and 84 two numbers at a time?

Yes, and it is the check this page prints. The GCF of 36 and 60 is 12, and the GCF of 12 and 84 is 12 — the same answer, because the greatest common factor is associative and does not depend on the order of the three numbers.

How does the GCF simplify the ratio 36 : 60 : 84?

Divide every term by the GCF 12: 36 ÷ 12 = 3, 60 ÷ 12 = 5 and 84 ÷ 12 = 7. So 36 : 60 : 84 in lowest terms is 3 : 5 : 7, and no whole number other than 1 divides all three of those.

See Also