FactorsAbout 120
120 is 1 × 2 × 3 × 4 × 5, written 5! and read "5 factorial". It has sixteen factors, more than any smaller number. A T20 innings is 120 balls.
Factor pairs of 120
A factor pair is two numbers that multiply to give 120: 1 × 120, 2 × 60, 3 × 40, 4 × 30, 5 × 24, 6 × 20, 8 × 15 and 10 × 12. Each pair is a way of arranging 120 things in equal rows, so 120 tiles can be laid out as a rectangle in 8 different shapes.
How to find the factors of 120 by hand
Divide 120 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 120: any larger factor has already turned up as the partner of a smaller one.
| Try | Divides 120? | Result |
|---|---|---|
| 1 | yes | 1 × 120 |
| 2 | yes | 2 × 60 |
| 3 | yes | 3 × 40 |
| 4 | yes | 4 × 30 |
| 5 | yes | 5 × 24 |
| 6 | yes | 6 × 20 |
| 7 | no | remainder 1 |
| 8 | yes | 8 × 15 |
| 9 | no | remainder 3 |
| 10 | yes | 10 × 12 |
The numbers that fail are 7 and 9. All the others give a pair.
Prime factorisation of 120
Divide by the smallest prime that goes in exactly, and repeat with the answer until you reach 1.
Reading down the left side: 120 = 2 × 2 × 2 × 3 × 5, which is written 23 × 3 × 5 with powers. Its prime factors are 2, 3 and 5.
Counting the factors without listing them
Add 1 to each power in the prime factorisation and multiply the results: (3 + 1) × (1 + 1) × (1 + 1) = 16. This works because a factor of 120 can use each prime anywhere from zero times up to its full power.
The 16 factors add up to 360. Leaving out 120 itself, the rest add up to 240, which is more than 120, so 120 is an abundant number. Of its factors, 4 are odd: 1, 3, 5 and 15.
Divisibility rules applied to 120
The quick tests agree with the factor list above.
- By 2: yes. The last digit is 0, which is even.
- By 3: yes. The digits add up to 1 + 2 + 0 = 3, which is a multiple of 3.
- By 4: yes. The last two digits make 20, and 20 ÷ 4 = 5.
- By 5: yes. The last digit is 0.
- By 6: yes. A number divides by 6 when it divides by both 2 and 3. Here both tests pass.
- By 8: yes. 120 ÷ 8 = 15.
- By 9: no. The digits add up to 3, which is not a multiple of 9.
- By 10: yes. The last digit is 0.
- By 11: no. Adding alternate digits from the right gives 1 and 2. Their difference is 1, which is not 0 or a multiple of 11.
HCF and LCM with 120
120 and 108. 120 = 23 × 3 × 5 and 108 = 22 × 33. 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 = 120 × 108.
120 and 125. 120 = 23 × 3 × 5 and 125 = 53. The primes they share, each with its smaller power, give the HCF: 5. Every prime with its larger power gives the LCM: 3000. By division: 125 = 120 × 1 + 5; 120 = 5 × 24 + 0, so the last divisor, 5, is the HCF. Check: 5 × 3000 = 15000 = 120 × 125.
Multiples of 120
Factors divide a number. Multiples are what you get by multiplying it. The first ten multiples of 120 are 120, 240, 360, 480, 600, 720, 840, 960, 1080 and 1200. A number has a fixed count of factors but endless multiples.
Common questions
What are the factors of 120?
The factors of 120 are 1, 2, 3, 4, 5, 6, 8, 10, 12, 15, 20, 24, 30, 40, 60 and 120. There are 16 of them.
What is the prime factorisation of 120?
120 = 2 × 2 × 2 × 3 × 5.
Is 120 a perfect square?
No. 10 × 10 = 100 and 11 × 11 = 121, and 120 lies between them.
What is the greatest factor of 120 other than itself?
60. Divide 120 by its smallest prime factor, 2, to get it.