FactorsAbout 108
108 is 4 × 27. In India it is well known as the number of beads on a japa mala.
Factor pairs of 108
A factor pair is two numbers that multiply to give 108: 1 × 108, 2 × 54, 3 × 36, 4 × 27, 6 × 18 and 9 × 12. Each pair is a way of arranging 108 things in equal rows, so 108 tiles can be laid out as a rectangle in 6 different shapes.
How to find the factors of 108 by hand
Divide 108 by 1, 2, 3 and so on. Each time the division is exact, you have found two factors at once: the divisor and the answer. You can stop at 10, because the next number, 11, gives 11 × 11 = 121, which is more than 108: any larger factor has already turned up as the partner of a smaller one.
| Try | Divides 108? | Result |
|---|---|---|
| 1 | yes | 1 × 108 |
| 2 | yes | 2 × 54 |
| 3 | yes | 3 × 36 |
| 4 | yes | 4 × 27 |
| 5 | no | remainder 3 |
| 6 | yes | 6 × 18 |
| 7 | no | remainder 3 |
| 8 | no | remainder 4 |
| 9 | yes | 9 × 12 |
| 10 | no | remainder 8 |
The numbers that fail are 5, 7, 8 and 10. All the others give a pair.
Prime factorisation of 108
Divide by the smallest prime that goes in exactly, and repeat with the answer until you reach 1.
Reading down the left side: 108 = 2 × 2 × 3 × 3 × 3, which is written 22 × 33 with powers. Its prime factors are 2 and 3.
Counting the factors without listing them
Add 1 to each power in the prime factorisation and multiply the results: (2 + 1) × (3 + 1) = 12. This works because a factor of 108 can use each prime anywhere from zero times up to its full power.
The 12 factors add up to 280. Leaving out 108 itself, the rest add up to 172, which is more than 108, so 108 is an abundant number. Of its factors, 4 are odd: 1, 3, 9 and 27.
Divisibility rules applied to 108
The quick tests agree with the factor list above.
- By 2: yes. The last digit is 8, which is even.
- By 3: yes. The digits add up to 1 + 0 + 8 = 9, which is a multiple of 3.
- By 4: yes. The last two digits make 8, and 8 ÷ 4 = 2.
- By 5: no. The last digit is 8, not 0 or 5.
- By 6: yes. A number divides by 6 when it divides by both 2 and 3. Here both tests pass.
- By 8: no. 108 ÷ 8 leaves remainder 4.
- By 9: yes. The digits add up to 9, which is a multiple of 9.
- By 10: no. The last digit is not 0.
- By 11: no. Adding alternate digits from the right gives 9 and 0. Their difference is 9, which is not 0 or a multiple of 11.
HCF and LCM with 108
108 and 100. 108 = 22 × 33 and 100 = 22 × 52. The primes they share, each with its smaller power, give the HCF: 4. Every prime with its larger power gives the LCM: 2700. By division: 108 = 100 × 1 + 8; 100 = 8 × 12 + 4; 8 = 4 × 2 + 0, so the last divisor, 4, is the HCF. Check: 4 × 2700 = 10800 = 108 × 100.
108 and 120. 108 = 22 × 33 and 120 = 23 × 3 × 5. The primes they share, each with its smaller power, give the HCF: 12. Every prime with its larger power gives the LCM: 1080. By division: 120 = 108 × 1 + 12; 108 = 12 × 9 + 0, so the last divisor, 12, is the HCF. Check: 12 × 1080 = 12960 = 108 × 120.
Multiples of 108
Factors divide a number. Multiples are what you get by multiplying it. The first ten multiples of 108 are 108, 216, 324, 432, 540, 648, 756, 864, 972 and 1080. A number has a fixed count of factors but endless multiples.
Common questions
What are the factors of 108?
The factors of 108 are 1, 2, 3, 4, 6, 9, 12, 18, 27, 36, 54 and 108. There are 12 of them.
What is the prime factorisation of 108?
108 = 2 × 2 × 3 × 3 × 3.
Is 108 a perfect square?
No. 10 × 10 = 100 and 11 × 11 = 121, and 108 lies between them.
What is the greatest factor of 108 other than itself?
54. Divide 108 by its smallest prime factor, 2, to get it.