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Factors of 90 are the numbers that divide 90 completely leaving no remainder. Factors of 90 are 1, 2, 3, 5, 6, 9, 10, 15, 18, 30, 45, and 90 in which 1 is considered the smallest factor and 90 is the greatest. The prime factorization of 90 is 2 × 3 × 3 × 5, thus the prime factors for the number 90 are 2, 3, and 5. Pair factors of 90 are the set of two numbers that give the product as 90 on multiplication with each other. The factors of 90 in pairs are (1, 90), (2, 45), (3, 30), (5, 18), (6, 15), and (9, 10). The sum of all the factors of 90 is 234.
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Key Terms: Factors of 90, Prime Factors, Prime Factorization, Pair Factors, Composite Number, Remainder, Division, Factor Tree
What are Factors of 90?
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Factors of 90 are the numbers that result in the original number 90 when multiplied by each other. All positive or negative numbers that divide 90 completely are called factors of 90. 90 is a composite number which means that it has more than two factors.
The number 90 has 12 factors in total which are 1, 2, 3, 5, 6, 9, 10, 15, 18, 30, 45, and 90.
- Positive Factors of 90: 1, 2, 3, 5, 6, 9, 10, 15, 18, 30, 45, and 90
- Negative Factors of 90: -1, -2, -3, -5, -6, -9, -10, -15, -18, -30, -45 and -90
Factors of 90
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How to Calculate Factors of 90?
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Factors of 90 can easily be calculated by dividing 90 by any natural number. If the number divides 90 completely and leaves zero as the remainder, then it is a factor of 90, else not. The factors of 90 can be calculated with the help of two methods which are as follows:
- Factors of 90 by Division Method
- Factors of 90 by Prime Factorization
Factors of 90 by Division Method
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The numbers that divide 90 perfectly without leaving a remainder are the factors of 90. By dividing 90 by multiple consecutive numbers, the factors of 90 can be determined using the division method.
- 90 ÷ 1 = 90 (Factor is 1; Remainder = 0)
- 90 ÷ 2 = 45 (Factor is 2; Remainder = 0)
- 90 ÷ 3 = 30 (Factor is 3; Remainder = 0)
- 90 ÷ 5 = 18 (Factor is 5; Remainder = 0)
- 90 ÷ 6 = 15 (Factor is 6; Remainder = 0)
- 90 ÷ 9 = 10 (Factor is 9; Remainder = 0)
- 90 ÷ 10 = 9 (Factor is 10; Remainder = 0)
- 90 ÷ 15 = 6 (Factor is 15; Remainder = 0)
- 90 ÷ 18 = 5 (Factor is 18; Remainder = 0)
- 90 ÷ 30 = 3 (Factor is 30; Remainder = 0)
- 90 ÷ 45 = 2 (Factor is 45; Remainder = 0)
- 90 ÷ 90 = 1 (Factor is 90; Remainder = 0)
Therefore, 1, 2, 3, 5, 6, 9, 10, 15, 18, 30, 45, and 90 are the factors of 90. Similarly, we observe that certain numbers, such as 4, 7, and 8, do not divide 90 completely and leave the remainder, so they are not factors of 90.
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Factors of 90 by Prime Factorization
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Prime factorization of 90 is the product of the prime factors of 90. Here is how to calculate the factors of 90 by prime factorization.
Factors of 90 by Prime Factorization
Start with the smallest prime number i.e. 2. Let's divide 90 by 2.
90 ÷ 2 = 45
45 is a composite number, hence it can be further divided by 5, which is a prime number.
45 ÷ 5 = 9
Now, 9 can be divided by 3.
9 ÷ 3 = 3
As a prime number, 3 can be further divided by 3 itself.
3 ÷ 3 = 1
Now, we can not proceed further, and thus, the prime factorization of 90 is 90 = 2 x 3 x 3 x 5. therefore, the prime factors of 90 are 2, 3, and 5.
Read More: Prime Number Formula
Factor Tree of 90
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In the factor tree method, the root of the factor tree is 90 since we need to determine the prime factors of 90. The pair factors of 90 are expressed as the branches of a tree. Here is how the factor tree method gives the factors of 90.
Factor Tree of 90
The factor tree comes to an end since 3 and 5 are prime numbers. Therefore, the prime factors of 90 are 2 x 3 x 3 x 5.
Factors of 90 in Pairs
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Pair factors of 90 are a pair of numbers that, when multiplied, produce a product as 90. The pair factors of 90 are:
| Factors of 90 | Pair Factors of 90 |
|---|---|
| 1 × 90 = 90 | (1, 90) |
| 2 × 45 = 90 | (2, 45) |
| 3 × 30 = 90 | (3,30) |
| 5 × 18 = 90 | (5,18) |
| 6 × 15 = 90 | (6, 15) |
| 9 × 10 = 90 | (9, 10) |
Thus, the factors of 90 in pairs are (1, 90), (2, 45), (3, 30), (5, 18), (15, 6), and (9, 10).
The pair factors of 90 can also be negative as the pair of negative numbers would also result in the positive value of 90. Thus, the negative pair factors of 90 are (-1, -90), (-2, -45), (-3, 30), (-5, 18), (-6, 15), and (-9, -10).
Solved Examples on Factors of 90
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Example 1: Rahul eats 3 mangoes every day. In how many days will he eat 90 mangoes?
Solution: It is given that
- Total Apples = 90
- Apples Per Day = 3
To find out the number of days he will take to finish 90 apples, we will divide 90 by 3
90 ÷ 3 = 30
The problem can also be solved by using the idea that (3, 30) are pair factors of 90. So, if 3 is one factor of 90, the other factor is 30. Thus, he will eat 90 apples in 30 days.
Example 2: List the Greatest Common Factor of 90 and 57.
Solution: We know that
- Factors of 90: 1, 2, 3, 5, 6, 9, 10, 15, 18, 30, 45, 90
- Factors of 57: 1, 3, 19, 57
The common factors of 90 and 57 are 1, and 3, thus, the Greatest Common Factor (GCF) of 90 and 57 is 3.
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| Related Topics | ||
|---|---|---|
| Co Prime Numbers | Perfect Numbers | Properties of LCM and HCF |
| HCF | LCM Formula | Greatest Common Divisor |
Things to Remember
- Factors of 90 are the numbers that divide 90 evenly with no remainder.
- There are 12 factors of the number 90.
- The factors of 90 are 1, 2, 3, 5, 6, 9, 10, 15, 18, 30, 45, and 90.
- The prime factorization of 90 is 2 x 3 x 3 x 5 with 2, 3, and 5 as the prime factors of 90.
- 90 has six pair factors in total which can be negative as well as positive.
- (1, 90), (2, 45), (3, 30), (5, 18), (6, 15), and (9, 10) are factors of 90 in pairs.
Sample Questions
Ques. Write down the common factors of 90 and 9. (2 Marks)
Ans. Writing all the factors of 90 and 9, we get
- Factors of 90 = 1, 2, 3, 5, 6, 9, 10, 15, 18, 30, 45, and 90.
- Factors of 9 = 1, 3, and 9.
The common factors of 90 and 9 are 1, 3, and 9.
Ques. Write down all the prime factors of 90. (2 Marks)
Ans. We know that
Factors of 90 = 1, 2, 3, 5, 6, 9, 10, 15, 18, 30, 45, and 90.
Therefore, the prime factors of 90 are 2, 3, and 5.
Ques. Aisha’s teacher informed her that one of the factors of 90 is (-18). What will the other factor be? (2 Marks)
Ans. Aisha must divide 90 by (-18) to obtain the other factor.
Since (-18) is one of the factors of 90.
Therefore, 90 ÷ (-18) = -5.
Thus, -5 is the other factor of 90.
Ques. Express the sum of 90 as two odd primes. (2 Marks)
Ans. The two odd prime numbers that add up to give 90 are 23 and 67.
90 = 23 + 67
Thus, the number 90 can be expressed as the sum of two odd prime numbers, 23 and 67.
Ques. Which is the greatest divisor between 90 and 60? (2 Marks)
Ans. The factors of 90 and 60 can be written as:
- 1, 2, 3, 5, 6, 9, 10, 15, 18, 30, 45, and 90 are the factors of 90.
- 1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30, and 60 are the factors of 60.
As shown, the greatest factor for 90 and 60 is 30. As a result, 30 is the greatest common divisor of 90 and 60.
Ques. Rishi has 90 chip packs. He intends to share it among his 15 family members. Find out how many packets each person will receive. (2 Marks)
Ans. We'll use the idea of factors of 90 here because Rishi has 90 packs of chips to divide among his 15 relatives:
Since 15 is a factor of 90, we get 6 when we divide 90 by 15.
90 ÷ 15 = 6
Rishi should therefore give each of his relatives 6 packets of chips.
Ques. Can we say 45 a factor of 90? (1 Mark)
Ans. Yes, 45 is a factor of 90. When we divide 90 by 45, we get 2 as the quotient and 0 as the remainder, which means that 45 is a factor of 90.
Ques. Diana enjoys five chocolates each day. How many days will she need to eat 90 chocolates? (2 Marks)
Ans. There are 90 chocolates in total.
We will divide 90 by 5 to get how many days it will take her to enjoy 90 chocolates. 90 ÷ 5 = 18
We know that (5, 18) are pair factors of 90. So, we may solve this using pair factors. Knowing that 5 is one of the factors of 90, the other factor one will be 18.
Therefore, she will eat 90 chocolates in 18 days.
Ques. List the positive and negative pair factors of 90. (2 Marks)
Ans. The positive and negative pair factors of 90 are as follows:
- Positive pair factors of 90: (1, 90), (2, 45), (3, 30), (5, 18), (15, 6), and (9, 10)
- Negative pair factors of 90: (-1, -90), (-2, -45), (-3, -30), (-5, -18), (-15, -6), and (-9, -10)
Ques. Find the sum of the factors of 90. (2 Marks)
Ans. We know that 1, 2, 3, 5, 6, 9, 10, 15, 18, 30, 45, and 90 are the factors of the number 90. When we add up all the numbers to determine the sum of the factors of 90, we get 234.
Therefore, the Sum of the factors of 90 is
1 + 2 + 3 + 5 + 6 + 9 + 10 + 15 + 18 + 30 + 45 + 90 = 234
Ques. State a few properties of factors. (2 Marks)
Ans. Some important properties of factors are
- 1 is considered the factor of every number.
- Every number is a factor of its own.
- There can be only a finite number of factors of a given number.
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