GCF of 9 and 12

Find the Greatest Common Factor (GCF) of 9 and 12 with a complete step-by-step solution. The GCF is the largest number that divides both numbers without leaving a remainder.

Answer: GCF(9, 12) = 3

Step-by-Step Solution

Step 1 Find the Factors of 9

To find the GCF of 9 and 12, we first need to find all factors of each number.

The factors of 9 are numbers that divide 9 evenly:

1 × 9 = 9, 3 × 3 = 9

Factors of 9: 1, 3, 9

Step 2 Find the Factors of 12

Now we find all factors of 12:

1 × 12 = 12, 2 × 6 = 12, 3 × 4 = 12

Factors of 12: 1, 2, 3, 4, 6, 12

Step 3 Find the Common Factors

The common factors are numbers that appear in both lists:

Factors of 9: 1, 3, 9

Factors of 12: 1, 2, 3, 4, 6, 12

Common factors: 1, 3

The Greatest Common Factor is the largest of these: 3

Step 4 Prime Factorization Method

We can also find the GCF using prime factorization:

9 = 3^2

12 = 2^2 × 3

Take the common prime factors with the lowest powers:

GCF = 3 = 3

Step 5 Euclidean Algorithm

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

12 = 1 × 9 + 3

9 = 3 × 3 + 0

When the remainder is 0, the GCF is the last divisor: 3

Step 6 Final Answer

GCF(9, 12) = 3

GCF of 9 and 12

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