What is the GCF of 72 and 60?

Understand the Problem

The question is asking for the greatest common factor (GCF) of the numbers 72 and 60. The GCF is the largest number that divides both of these numbers without leaving a remainder. To find the GCF, we can list the factors of both numbers or use the prime factorization method.

Answer

$12$

Steps to Solve

Find the prime factorization of 72

To find the prime factorization of 72, we divide it by the smallest prime number (2) until we can no longer divide.

$$ 72 \div 2 = 36 \ 36 \div 2 = 18 \ 18 \div 2 = 9 \ 9 \div 3 = 3 \ 3 \div 3 = 1 $$

The prime factorization of 72 is $2^3 \times 3^2$.

Find the prime factorization of 60

Now, we do the same for 60:

$$ 60 \div 2 = 30 \ 30 \div 2 = 15 \ 15 \div 3 = 5 \ 5 \div 5 = 1 $$

The prime factorization of 60 is $2^2 \times 3^1 \times 5^1$.

Identify common prime factors

Next, we look for the common prime factors in the factorizations of 72 and 60. The common factors are:

For $2$: the minimum power is $2^2$ (from 60)
For $3$: the minimum power is $3^1$ (from 60)

Calculate the GCF

Now, we multiply the common prime factors with their minimum powers:

$$ GCF = 2^2 \times 3^1 = 4 \times 3 = 12 $$

The greatest common factor of 72 and 60 is 12.

The greatest common factor (GCF) of 72 and 60 is $12$.

More Information

The GCF is useful in simplifying fractions, finding common denominators, and solving problems involving ratios. It's also interesting to know that the GCF can help determine how to split a set of items into equal groups without leftovers.

Tips

Forgetting to include all prime factors when determining factorization.
Not checking for the minimum power of the common factors.
Not recognizing that both numbers must be divided by the GCF without leaving a remainder.

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