Factors of 90 are the numbers, which gives the result as 90 when multiplied together in pair of two. The factors of 90 can be positive or negative, but they cannot be in the fraction or decimal form. For example, the pair factors of 90 can be (1, 90) or (-1, -90). If we multiply the pair of negative factors, such as multiplying -1 and -90, we will get the original number 90. The factors of a number can be found using the division method as well as the prime factorization method. In this article, we will learn what are the factors of 90, pair factors and the prime factors of 90 using the two different methods and many solved examples.

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## What are the Factors of 90?

The factors of 90 are the numbers, which are multiplied in pairs resulting in an original number 90. As the number is an even composite number, it has many factors other than 1 and 90. The factors of 90 can be positive or negative. Thus, the positive factors of 90 are 1, 2, 3, 5, 6, 9, 10, 15, 18, 30, 45, and 90.

 Factors of 90: 1, 2, 3, 5, 6, 9, 10, 15, 18, 30, 45, and 90Prime Factorization of 90: 3×3×5×2 or 32 × 5 × 2

## Pair Factors of 90

The pair of numbers which are multiplied together, resulting in an original number 90 is the pair factors of 90. The positive and the negative pair factors of 90 are given below:

Positive Pair Factors of 90:

 Positive Factors of 90 Positive Pair Factors of 90 1 × 90 (1, 90) 2 × 45 (2, 45) 3 × 30 (3, 30) 5 × 18 (5, 18) 6 × 15 (6, 15) 9 × 10 (9, 10)

Negative Pair Factors of 90:

 Negative Factors of 90 Negative Pair Factors of 90 -1 × -90 (-1, -90) -2 × -45 (-2, -45) -3 × -30 (-3, -30) -5 × -18 (-5, -18) -6 × -15 (-6, -15) -9 × -10 (-9, -10)

## Factors of 90 by Division Method

In the division method, the factors of 90 can be found by dividing 90 by different consecutive integers. If the integer divides 90 exactly without leaving a remainder, then the integers are the factors of 90. Now, start dividing 90 by different integer numbers.

90/1 = 90 (Factor is 1 and remainder is 0)90/2 = 45 (Factor is 2 and remainder is 0)90/3 = 30 (Factor is 3 and remainder is 0)90/5 = 18 (Factor is 5 and remainder is 0)90/6 = 15 (Factor is 6 and remainder is 0)90/9 = 10 (Factor is 9 and remainder is 0)90/10 = 9 (Factor is 10 and remainder is 0)90/15 = 6 (Factor is 15 and remainder is 0)90/18 = 5 (Factor is 18 and remainder is 0)90/30 = 3 (Factor is 30 and remainder is 0)90/45 = 2 (Factor is 45 and remainder is 0)90/90 = 1 (Factor is 90 and remainder is 0)

If we divide 90 by numbers other than 1, 2, 3, 5, 6, 9, 10, 15, 18, 30, 45 and 90, we will get a remainder. Hence, the factors of 90 are 1, 2, 3, 5, 6, 9, 10, 15, 18, 30, 45, and 90.

## Prime Factorization of 90

The number 90 is a composite number. Now, the prime factors of 90 can be found as shown below:

The first step is to divide the number 90 with the smallest prime factor, i.e. 2 and divide the output again by 2 till you get a fraction or odd number.

90 ÷ 2 = 45;

Divide 45 by 2.

45 ÷ 2 = 22.5; fraction cannot be a factor.

Therefore, moving to the next prime numbers, 3, 5, 7 and so on.

Divide 45 by 3.

45 ÷ 3 = 15

15 ÷ 3 = 5

Now, we know 5 is a prime number and it has only two factors; 1 and 5. Therefore, we cannot continue with the division method further.So, the prime factors of 90 are 2 × 3 × 3 × 5 or 2 × 32 × 5, where 2, 3 and 5 are the prime numbers.

### Sum of Factors of 90

We know that the factors of the number 90 are 1, 2, 3, 5, 6, 9, 10, 15, 18, 30, 45, 90. Now, to find the sum of the factors, add all the numbers, you will get a total of 234.

Sum of factors of 90 = 1 + 2 + 3 + 5 + 6 + 9 + 10 + 15 + 18 + 30 + 45 + 90

Sum of factors of 90 = 234

If you exclude the number 90, you will get 144, i.e. 234 – 90 = 144

### Examples

Example 1:

Find the common factors of 90 and 91.

Solution:

The factors of 90 are 1, 2, 3, 5, 6, 9, 10, 15, 18, 30, 45, and 90.

The factors of 91 are 1, 7, 13, and 91.

Therefore, the common factor of 90 and 91 is 1.

Example 2:

Find the common factors of 90 and 89.

Solution:

Factors of 90 = 1, 2, 3, 5, 6, 9, 10, 15, 18, 30, 45, and 90.

Factors of 89 = 1 and 89.

As 89 is a prime number, the common factor of 90 and 89 is 1 only.

Example 3:

Find the common factors of 90 and 100.

Solution:

The factors of 90 are 1, 2, 3, 5, 6, 9, 10, 15, 18, 30, 45, and 90.

The factors of 100 are 1, 2, 4, 5, 10, 20, 25, 50, and 100.

Hence, the common factors of 90 and 100 are 1, 2, 5 and 10.

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