180, a positive integer with more divisors than any other smaller positive integer, is also a highly composite number. 180 is a number which is refactorable. In this lesson, we will calculate the factors of 180, prime factors of 180, and factors of 180 in pairs along with solved examples for a better understanding.
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| 1. | What Are the Factors of 180? |
| 2. | How to Calculate the Factors of 180? |
| 3. | Factors of 180 by Prime Factorization |
| 4. | Factors of 180 in Pairs |
| 5. | Important Notes |
| 6. | FAQs on Factors of 180 |
What Are the Factors of 180?
The number 180 is an even composite number. As it is even, it will have 2 as its factor. To understand why it is composite, let"s recall the definition of a composite number. A number is said to be composite if it has more than two factors. Consider the number 19. It has only two factors, 1 and 19. So, it"s prime. Now, let"s take the case of 60. The factors of 60 are 1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30, and 60. There are more than two factors of 60, hence it"s composite. Let"s come back to 180. The factors of 180 are all the integers that 180 can be divided into.
How to Calculate the Factors of 180?
We can use different methods like prime factorization and the division method to calculate the factors of 180. In prime factorization, we express 180 as a product of its prime factors and in the division method, we see what numbers divide 180 exactly without a remainder.
Factors of 180 by Prime Factorization
The number 180 is divided by the smallest prime number which divides 180 exactly, i.e., it leaves a remainder of 0. The quotient is then divided by the smallest or second smallest prime number and the process continues till the quotient gets undividable. Let us divide 180 by the prime number 2.
180 ÷ 2 = 90Now we need to divide the quotient 90 by the next least prime number.
90 ÷ 2 = 45Now, 45 is a odd number so it is not divisible by 2, but it is divisible by 3.
45 ÷ 3 = 15Again 15 is divisible by 3.
15 ÷ 3 = 55 is a prime number we cannot divide further. Therefore, the prime factors of 180 are 2, 3, and 5 only. Exponentially it can be written as 180 = 22 x 32 x 5
Explore factors using illustrations and interactive examples.
Factors of 180 in Pairs
The pairs of numbers which give 180 when multiplied are known as factor pairs of 180. The following are the factors of 180 in pairs or factor pairs of 180 can be given as
| The product as 180 | Pair factor |
| 1 × 180 = 180 | (1, 180) |
| 2 × 90 = 180 | (2, 90) |
| 3 × 60 = 180 | (3, 60) |
| 4 × 45 = 180 | (4, 45) |
| 5 × 36 = 180 | (5, 36) |
| 6 × 30 = 180 | (6, 30) |
| 9 × 20 = 180 | (9, 20) |
| 10 × 18 = 180 | (10, 18) |
| 12 × 15 = 180 | (12, 15) |
| 15 × 12 = 180 | (15, 12) |
| 18 × 10 = 180 | (18, 10) |
| 20 × 9 = 180 | (20, 9) |
| 30 × 6 = 180 | (30, 6) |
| 36 × 5 = 180 | (36, 5) |
| 45 × 4 = 180 | (45, 4) |
| 60 × 3 = 180 | (60, 3) |
| 90 × 2 = 180 | (90, 2) |
| 180 × 1 = 180 | (180, 1) |
Observe in the table above, after 12 × 15, the factors start repeating except the order. So, it is enough to find factors until (12,15).
If we consider negative integers, then both the numbers in the pair factors will be negative.36 is positive and - ve (×) - ve = +ve.So, we can have factor pairs of 180 as (-1,-180) ; (-2,-90); (-4,-45); (-5,-36) ; (-6,-30) ; (-9, -20); (-10,-18) ; (-9, -20).
Important Notes:
The numbers which we multiply to get 180 are the factors of 180.Factors of 180 are 1, 2, 3, 4, 5, 6, 9, 10, 12, 15, 18, 20, 30, 36, 45, 60, 90, and 180.Factor pairs of 180 are (1,180) (2, 90) (3, 60) (4,45) (5, 36) (6, 30) (9, 20) (10, 18 ) and (12, 15).
Challenging Questions
Do you know 180 is a refactorable number?Do you know 180 is an abundant number, with its proper divisors summing up to 366?Example 1: Can you help Lisa list the factors of 180 and find the factor pairs?
Solution:
The factors of 180 are the numbers that divide without any remainder.They are 1, 2, 3, 4, 5, 6, 9, 10, 12, 15, 18, 20, 30, 36, 45, 60, 90, and 180.Factors pairs are the pairs of two numbers which, when multiplied, give 180.
1 × 180 = 1802 × 90 = 1803 × 60 = 1804 × 45 = 1805 × 36 = 1806 × 30 = 1809 × 20 = 18010 × 18 = 180Factors of 180 are 1, 2, 3, 4, 5, 6, 9, 10, 12, 15, 18, 20, 30, 36, 45, 60, 90, and 180.Factor pairs of 180 are (1,180), (2,90), (3, 60), (4,45), (5, 36), (6, 30), (9, 20), (10, 18), and (12, 15).
Example 2: Joel has to list the factors of 180 and the factors of 120 and find the common factors between them. Can you help him do it?
Solution:
Factors of 120 are 1, 2, 3, 4, 5, 6, 8, 10, 12, 15, 20, 24, 30, 40, 60, and 120.Factors of 180 are 1, 2, 3, 4, 5, 6, 9, 10, 12, 15, 18, 20, 30, 36, 45, 60, 90, and 180.We can see that 1, 2, 5, 10, 12, 15, 20, 30, and 60 are the common factors of 120 and 180.Therefore, common factors of 120 and 180 are 1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30, and 60.
Example 3: The area of a rectangle is 180 square inches. List all the possible combinations possible for length and breadth.
See more: Which Fraction Is 5/16 Bigger Than 3/8 And 5/16, Which Fraction Is Greater?
Solution:
Area of rectangle = length × breadthGiven the area is 180 square inches, the possible length and breadth are the factors of 180.There are 18 possible combinations.We can swap the values of length and breadth.
| Length | Breadth |
| 1 in | 180 in |
| 2 in | 90 in |
| 3 in | 60 in |
| 4 in | 45 in |
| 5 in | 36 in |
| 6 in | 30 in |
| 9 in | 20 in |
| 10 in | 18 in |
| 12 in | 15 in |
| 15 in | 12 in |
| 18 in | 10 in |
| 20 in | 9 in |
| 30 in | 6 in |
| 36 in | 5 in |
| 45 in | 4 in |
| 60 in | 3 in |
| 90 in | 2 in |
| 180 in | 1 in |
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