What are all the factors, the prime factorization, and factor pairs of 72?

Factors are fundamental components in the world of mathematics. These are the numbers we multiply together to get another number, and in this case, we will explore 72 as our illustrative example. Understanding the factors of 72 sheds light on the structure and properties of this number within various mathematical contexts.

The concept of factors applies to the number 72 in a way that is both simple and profound. A factor of 72 is any number that can evenly divide 72, leaving no remainder. Below is a visual representation of these factors, including their pairings:

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

As you can see, factors of 72 include: 1, 2, 3, 4, 6, 8, 9, 12, 18, 24, 36, and 72.

Finding the factors of 72 involves a straightforward division method. Here’s how you can find them:

To determine if a number is a factor of 72, we’ll start with the smallest natural number, 1, and move to higher numbers to see which ones divide evenly without leaving a fraction. Let’s look at the division of 72 by several integers:

• 72 ÷ 1 = 72
• 72 ÷ 2 = 36
• 72 ÷ 3 = 24
• 72 ÷ 4 = 18
• 72 ÷ 6 = 12
• 72 ÷ 8 = 9
• 72 ÷ 9 = 8
• 72 ÷ 12 = 6
• 72 ÷ 18 = 4
• 72 ÷ 24 = 3
• 72 ÷ 36 = 2
• 72 ÷ 72 = 1

If any number not listed there is divided into 72, the result will not be a whole number, indicating that it is not a factor of 72.

Pair factors of 72 are two numbers that, when multiplied together, result in 72. Here is a complete list of both positive and negative pair factors:

The prime factorization of 72 is the process of breaking it down into prime numbers that multiply together to result in 72. Follow these steps to create a factor tree of 72:

• Start with two factors, like 2 and 36.
• Since 2 is a prime number, it is one of the prime factors.
• Continue breaking down 36 into its prime factors, which are also 2 and 18.
• Eventually, 72 is broken down into 2 × 2 × 2 × 3 × 3 as the prime factors.

The prime factorization of 72 can be written as 2³ × 3².

When discussing the factors of 72, there are a few important points to keep in mind:

• There are 12 factors of 72 in total.
• The prime factors of 72 are 2 and 3.
• The pair factors come in both positive and negative.

The sum of all factors of 72 is 195 (1+2+3+4+6+8+9+12+18+24+36+72).

1. List all the factors of 72 that are prime numbers.

2. What are the pair factors of 72 that multiply to give a product of 72?

3. If the prime factors of 72 are subtracted (2 and 3), what number remains?

1. The prime factors of 72 are 2 and 3.

2. The pair factors of 72 are (1, 72), (2, 36), (3, 24), (4, 18), (6, 12), and (8, 9).

3. Subtracting the prime factors from 72 leaves us with 67 (72 – 2 – 3 = 67).