GCF of 72 and 84 is the largest possible number that divides 72 and 84 exactly without any remainder. The factors of 72 and 84 are 1, 2, 3, 4, 6, 8, 9, 12, 18, 24, 36, 72 and 1, 2, 3, 4, 6, 7, 12, 14, 21, 28, 42, 84 respectively. There are 3 commonly used methods to find the GCF of 72 and 84 - prime factorization, Euclidean algorithm, and long division.

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Answer: GCF of 72 and 84 is 12.

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Explanation:

The GCF of two non-zero integers, x(72) and y(84), is the greatest positive integer m(12) that divides both x(72) and y(84) without any remainder.


The methods to find the GCF of 72 and 84 are explained below.

Prime Factorization MethodUsing Euclid's AlgorithmLong Division Method

GCF of 72 and 84 by Prime Factorization

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Prime factorization of 72 and 84 is (2 × 2 × 2 × 3 × 3) and (2 × 2 × 3 × 7) respectively. As visible, 72 and 84 have common prime factors. Hence, the GCF of 72 and 84 is 2 × 2 × 3 = 12.

GCF of 72 and 84 by Euclidean Algorithm

As per the Euclidean Algorithm, GCF(X, Y) = GCF(Y, X mod Y)where X > Y and mod is the modulo operator.

Here X = 84 and Y = 72

GCF(84, 72) = GCF(72, 84 mod 72) = GCF(72, 12)GCF(72, 12) = GCF(12, 72 mod 12) = GCF(12, 0)GCF(12, 0) = 12 (∵ GCF(X, 0) = |X|, where X ≠ 0)

Therefore, the value of GCF of 72 and 84 is 12.

GCF of 72 and 84 by Long Division

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GCF of 72 and 84 is the divisor that we get when the remainder becomes 0 after doing long division repeatedly.

Step 2: Since the remainder ≠ 0, we will divide the divisor of step 1 (72) by the remainder (12).Step 3: Repeat this process until the remainder = 0.

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The corresponding divisor (12) is the GCF of 72 and 84.