What are all the factors, the prime factorization, and factor pairs of 90?

Factors are the fundamental building blocks in the world of numbers, integral to understanding the composition of integers. We can think of factors as the numbers we multiply together to get another number. For instance, with 90 as our illustrative example, analyzing the factors of 90 helps in revealing its internal structure and mathematical properties.

Factors of 90 are the whole numbers that can divide 90 without leaving any remainder. When paired and multiplied, these factors give a product of 90. Here is a visual representation and the actual list of factors of 90:

• 1 x 90 = 90
• 2 x 45 = 90
• 3 x 30 = 90
• 5 x 18 = 90
• 6 x 15 = 90
• 9 x 10 = 90

We can find the factors of 90 by utilizing the division method, which involves dividing 90 by integers that result in a whole number quotient. Each divisor that gives a whole number quotient is a factor of 90:

• 90 ÷ 1 = 90
• 90 ÷ 2 = 45
• 90 ÷ 3 = 30
• 90 ÷ 5 = 18
• 90 ÷ 6 = 15
• 90 ÷ 9 = 10
• 90 ÷ 10 = 9

Apart from these, dividing 90 by any other number will result in a non-whole quotient, indicating that the number is not a factor. For instance, 90 ÷ 4 = 22.5, thus 4 is not a factor of 90.

Consider the concept of pair factors as a way to showcase two integers that when multiplied yield 90.

The prime factorization of 90 can be found by breaking it down into its prime components. A prime factor tree visualizes this breakdown:

• 90 ÷ 2 = 45
• 45 ÷ 3 = 15
• 15 ÷ 3 = 5
• 5 ÷ 5 = 1

In conclusion, the prime factors of 90 are 2, 3, and 5; hence the prime factorization of 90 is 2 x 3^2 x 5.

Here are the key points to remember about the factors of 90:

• Total number of factors: 12
• Prime factors: 2, 3, and 5
• Pair factors: (1, 90), (2, 45), (3, 30), (5, 18), (6,15), and (9, 10)
• Sum of all factors: 234
• Understanding the factors of 90 can be useful in various mathematical contexts.

Put your knowledge to the test with these exercises related to the factors of 90:

1. List all the prime factors of 90.
2. What two pair factors of 90 multiply to give a product of 90?
3. Find the sum of the first three factors of 90.

Here are the solutions to the exercises on factors of 90:

• Prime factors of 90: 2, 3, 5
• Two pair factors of 90 that multiply to give 90: (3, 30) and (5, 18)
• Sum of the first three factors of 90: 1 + 2 + 3 = 6