GCF of 72 and 96

Find the Greatest Common Factor (GCF) of 72 and 96 with a complete step-by-step solution. The GCF is the largest number that divides both numbers without leaving a remainder.

Answer: GCF(72, 96) = 24

Step-by-Step Solution

Step 1 Find the Factors of 72

To find the GCF of 72 and 96, we first need to find all factors of each number.

The factors of 72 are numbers that divide 72 evenly:

1 × 72 = 72, 2 × 36 = 72, 3 × 24 = 72, 4 × 18 = 72, 6 × 12 = 72, 8 × 9 = 72

Factors of 72: 1, 2, 3, 4, 6, 8, 9, 12, 18, 24, 36, 72

Step 2 Find the Factors of 96

Now we find all factors of 96:

1 × 96 = 96, 2 × 48 = 96, 3 × 32 = 96, 4 × 24 = 96, 6 × 16 = 96, 8 × 12 = 96

Factors of 96: 1, 2, 3, 4, 6, 8, 12, 16, 24, 32, 48, 96

Step 3 Find the Common Factors

The common factors are numbers that appear in both lists:

Factors of 72: 1, 2, 3, 4, 6, 8, 9, 12, 18, 24, 36, 72

Factors of 96: 1, 2, 3, 4, 6, 8, 12, 16, 24, 32, 48, 96

Common factors: 1, 2, 3, 4, 6, 8, 12, 24

The Greatest Common Factor is the largest of these: 24

Step 4 Prime Factorization Method

We can also find the GCF using prime factorization:

72 = 2^3 × 3^2

96 = 2^5 × 3

Take the common prime factors with the lowest powers:

GCF = 2 × 2 × 2 × 3 = 24

Step 5 Euclidean Algorithm

The Euclidean algorithm is a fast way to find the GCF using repeated division:

96 = 1 × 72 + 24

72 = 3 × 24 + 0

When the remainder is 0, the GCF is the last divisor: 24

Step 6 Final Answer

GCF(72, 96) = 24

GCF of 72 and 96

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