The greatest common factor (GCF) of two or more numbers is the largest number that divides evenly into each of the numbers. To find the GCF of 64 and 72, we need to factor each number into its prime factors and look for common factors. Once we determine the common factors, the greatest of these will be the GCF of 64 and 72. Knowing how to find the GCF is an important skill in mathematics, as the GCF comes up often when simplifying fractions or solving certain types of word problems. In this article, we will walk through the full process of finding the GCF of 64 and 72. We will also look at some examples of how the GCF is used in practical applications.

## Factoring 64 and 72 into Prime Factors

To find the GCF of two numbers, we first need to break each number down into its unique prime factors. A prime number is a number greater than 1 that has no positive divisors other than 1 and itself. Some examples of prime numbers are 2, 3, 5, 7, and 11. Prime factorization involves continuously dividing a composite number by prime numbers to determine the primes that, when multiplied together, equal the original number.

Let’s start by prime factorizing 64:

Number | Prime Factorization |
---|---|

64 | 2 x 2 x 2 x 2 x 2 x 2 |

We can see that 64 equals 2^{6}. The number 64 contains only the prime factor 2.

Now let’s prime factor 72:

Number | Prime Factorization |
---|---|

72 | 2 x 2 x 2 x 3 x 3 |

72 equals 2^{3} x 3^{2}. The prime factors of 72 are 2 and 3.

By breaking down 64 and 72 into their unique prime factors, we can more easily identify their common factors.

## Identifying Common Factors

Now that we have the prime factorizations of 64 and 72, we can look for common factors:

- 64 = 2 x 2 x 2 x 2 x 2 x 2 = 2
^{6} - 72 = 2 x 2 x 2 x 3 x 3 = 2
^{3}x 3^{2}

We can see that both 64 and 72 are divisible by 2. Specifically, 64 is divisible by 2^{6} and 72 is divisible by 2^{3}. This means that 2^{3} is a common factor of both numbers.

In addition, neither 64 nor 72 is divisible by any common prime factors other than 2. Therefore, the greatest common factor of 64 and 72 is 2^{3}, which equals 8.

## GCF of 64 and 72

Based on the prime factorizations and common factors identified above, we can conclude:

The GCF of 64 and 72 is **8**.

This makes sense when we think about it conceptually. Both 64 and 72 are divisible by 8 with no remainder (64/8 = 8 and 72/8 = 9). And 8 is the greatest number by which they are both divisible, so it is their GCF.

## Applications of the GCF

Now that we have found the GCF of 64 and 72, let’s look at some examples of how the GCF can be used:

### Simplifying Fractions

We can use the GCF to reduce fractions to their simplest forms. For example, consider the fraction:

64/72

Since the GCF of 64 and 72 is 8, we can divide both the numerator and denominator by 8:

64/72 = (64/8)/(72/8) = 8/9

Therefore, 64/72 simplifies to 8/9. The GCF allowed us to reduce the original fraction.

### Word Problems

The GCF can also be useful for solving certain types of word problems. Here is an example:

Jessica has 64 pencils and 72 pens. She wants to distribute them evenly into gift bags with the same number of pencils and pens in each bag. What is the greatest number of gift bags she can make?

To solve this, we first find the GCF of 64 and 72, which is 8. This tells us the greatest number we can divide both 64 and 72 by to get whole numbers for the quantities in each gift bag is 8. Therefore, the greatest number of gift bags Jessica can make is 64/8 = 72/8 = 8 bags.

## Conclusion

Finding the greatest common factor (GCF) involves breaking down the numbers into prime factors, identifying common factors, and determining the greatest of these. For the numbers 64 and 72, the prime factorizations are 2^{6} and 2^{3} x 3^{2}, respectively. The only common prime factor is 2^{3} = 8, so this is their GCF. Knowledge of how to find the GCF is useful for simplifying fractions, solving certain word problems, and other mathematical applications. With some practice, finding the GCF becomes a simple process.