GCF for 12, 15 and 21

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

What's the Greatest Common Factor (GCF) of 12, 15 and 21?

Answer

GCF of 12, 15 and 21 is
3

(Three)

3 is the largest positive integer that divides 12, 15 and 21 without a remainder — every other common divisor of the three divides it too.

Finding the GCF of 12, 15 and 21 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 12: 2 × 2 × 3 (exponent form: 22 × 3)

All Prime Factors of 15: 3 × 5 (exponent form: 3 × 5)

All Prime Factors of 21: 3 × 7 (exponent form: 3 × 7)

Prime factor121521Lowest power
222———
33333
5—5——
7——7—

GCF = 3 = 3

Checking the GCF of 12, 15 and 21 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(12, 15) = 3

GCF(3, 21) = 3

GCF(12, 15, 21) = 3

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

12 ÷ 3 = 4

15 ÷ 3 = 5

21 ÷ 3 = 7

4, 5 and 7 have no common factor left — that is what makes 3 the greatest one.

Related Calculations

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 12, 15 and 21? (The answer is: 3). Enter the first number (e.g. '12'), the second (e.g. '15') and the third (e.g. '21'), 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
612153
712151
812151
912153
1012151
1112151
1212153
1213151
1214151
1215153
1215161
1215171
1215183
1215191
1215201
1215213
1215221
1215231
1215243
1215251
1215261
1215273
1215281
1215291
1215303
1215311
1215321
1215333
1215341
1215351

FAQ

What's the Greatest Common Factor (GCF) of 12, 15 and 21?

GCF of 12, 15 and 21 is 3

What prime factors do 12, 15 and 21 share?

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

Can you get the GCF of 12, 15 and 21 two numbers at a time?

Yes, and it is the check this page prints. The GCF of 12 and 15 is 3, and the GCF of 3 and 21 is 3 — 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 12 : 15 : 21?

Divide every term by the GCF 3: 12 ÷ 3 = 4, 15 ÷ 3 = 5 and 21 ÷ 3 = 7. So 12 : 15 : 21 in lowest terms is 4 : 5 : 7, and no whole number other than 1 divides all three of those.

See Also