What Are The Factors Of 108

The factors of 108 are the numbers that divide 108 exactly, leaving no remainder. Understanding factors is fundamental in mathematics, appearing in areas like simplifying fractions, finding greatest common divisors, and solving algebraic equations. This article provides a comprehensive exploration of the factors of 108, explaining how to find them systematically and highlighting their significance.

Introduction

A factor of a number is an integer that divides that number evenly, without leaving a remainder. Take this case: 3 is a factor of 108 because 108 ÷ 3 = 36, a whole number. Conversely, 5 is not a factor because 108 ÷ 5 = 21.6, which isn't an integer. So naturally, identifying all factors of a number like 108 is a crucial skill in arithmetic and algebra. This article will detail the complete list of factors for 108 and explain the methods used to find them efficiently.

How to Find the Factors of 108

Finding all factors of a number involves a systematic approach. Here's how to do it:

1. Start with 1 and the Number Itself: Every number is divisible by 1 and itself. Which means, 1 and 108 are always factors.
2. Check Divisibility by 2: 108 is even (ends in 0), so it's divisible by 2. 108 ÷ 2 = 54. Thus, 2 and 54 are factors.
3. Check Divisibility by 3: Sum the digits: 1 + 0 + 8 = 9, which is divisible by 3. So, 108 is divisible by 3. 108 ÷ 3 = 36. So, 3 and 36 are factors.
4. Check Divisibility by 4: Look at the last two digits (08). 08 ÷ 4 = 2, an integer. So, 108 is divisible by 4. 108 ÷ 4 = 27. Thus, 4 and 27 are factors.
5. Check Divisibility by 5: Numbers ending in 0 or 5 are divisible by 5. 108 ends in 8, so it's not divisible by 5. 5 is not a factor.
6. Check Divisibility by 6: A number is divisible by 6 if it's divisible by both 2 and 3. Since 108 is divisible by both 2 and 3 (as established), it's divisible by 6. 108 ÷ 6 = 18. So, 6 and 18 are factors.
7. Check Divisibility by 7: Perform the division: 108 ÷ 7 ≈ 15.428..., not an integer. 7 is not a factor.
8. Check Divisibility by 8: The last three digits (108) ÷ 8 = 13.5, not an integer. 8 is not a factor.
9. Check Divisibility by 9: Sum the digits: 1 + 0 + 8 = 9, which is divisible by 9. That's why, 108 is divisible by 9. 108 ÷ 9 = 12. So, 9 and 12 are factors.
10. Check Divisibility by 10: Numbers ending in 0 are divisible by 10. 108 ends in 8, so it's not divisible by 10. 10 is not a factor.
11. Check Divisibility by 11: Alternating sum: 1 - 0 + 8 = 9. 9 is not divisible by 11. 11 is not a factor.
12. Check Divisibility by 12: Since 12 is a factor (from step 9), and we've already established 12 is a factor, we don't need to check it again. We continue checking higher numbers.
13. Check Divisibility by 13: 108 ÷ 13 ≈ 8.307..., not an integer. 13 is not a factor.
14. Check Divisibility by 14: A number divisible by 14 must be divisible by both 2 and 7. It's divisible by 2, but not by 7 (step 7). That's why, 14 is not a factor.
15. Check Divisibility by 15: A number divisible by 15 must be divisible by both 3 and 5. It's divisible by 3, but not by 5 (step 5). Which means, 15 is not a factor.
16. Check Divisibility by 16: The last three digits (108) ÷ 16 = 6.75, not an integer. 16 is not a factor.
17. Check Divisibility by 17: 108 ÷ 17 ≈ 6.353..., not an integer. 17 is not a factor.
18. Check Divisibility by 18: Since 18 is already a factor (from step 6), and we've established it, we don't need to check it again. We continue checking higher numbers.
19. Check Divisibility by 19: 108 ÷ 19 ≈ 5.684..., not an integer. 19 is not a factor.
20. Check Divisibility by 20: A number divisible by 20 must end in 00, 20, 40, 60, or 80. 108 ends in 08, so it's not divisible by 20. 20 is not a factor.
21. Check Divisibility by 21: A number divisible by 21 must be divisible by both 3 and 7. It's divisible by 3, but not by 7 (step 7). Which means, 21 is not a factor.
22. Check Divisibility by 22: A number divisible by 22 must be divisible by both 2 and 11. It's divisible by 2, but not by 11 (step 11). Which means, 22 is not a factor.
23. Check Divisibility by 23: 108 ÷ 23 ≈ 4.695..., not an integer. 23 is not a factor.
24. Check Divisibility by 24: A number divisible