What is the greatest common factor of 40 and 72?

Understand the Problem

The question is asking for the greatest common factor (GCF) of the numbers 40 and 72. The GCF is the largest integer that divides both numbers without leaving a remainder. We will approach this by finding the prime factors of both numbers and identifying the common factors.

Answer

The GCF of 40 and 72 is $8$.

Steps to Solve

Find the prime factorization of 40

To begin, we need to determine the prime factors of 40.

We can do this by dividing 40 by the smallest prime number, which is 2:

$$ 40 \div 2 = 20 $$ $$ 20 \div 2 = 10 $$ $$ 10 \div 2 = 5 $$

Since 5 is a prime number, we can stop here.

So, the prime factorization of 40 is:

$$ 40 = 2^3 \times 5^1 $$

Find the prime factorization of 72

Next, we will find the prime factors of 72 using the same method.

Starting with the smallest prime number:

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

The prime factorization of 72 is:

$$ 72 = 2^3 \times 3^2 $$

Identify common factors

Now, we need to find the common prime factors of both numbers.

From the factorizations, we have:

For 40: $2^3 \times 5^1$
For 72: $2^3 \times 3^2$

The common prime factor is $2$, and the lowest exponent in both factorizations for the common prime factor $2$ is $3$.

Calculate the GCF

To find the GCF, we multiply the common prime factors raised to their lowest exponents:

$$ GCF = 2^3 = 8 $$

The greatest common factor (GCF) of 40 and 72 is $8$.

More Information

The greatest common factor helps in simplifying fractions and finding common denominators. Knowing how to find the GCF is useful in many areas of math, including algebra and number theory.

Tips

Common mistakes include:

Not fully factorizing the numbers.
Misidentifying the lowest exponent of the common prime factor.
Forgetting to multiply the common prime factors to find the GCF.

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