FactorsAbout 60
An hour has 60 minutes and a minute has 60 seconds. 60 is the smallest number that divides evenly by 1, 2, 3, 4, 5 and 6, which makes it very handy for measuring time.
Factor pairs of 60
A factor pair is two numbers that multiply to give 60: 1 × 60, 2 × 30, 3 × 20, 4 × 15, 5 × 12 and 6 × 10. Each pair is a way of arranging 60 things in equal rows, so 60 tiles can be laid out as a rectangle in 6 different shapes.
How to find the factors of 60 by hand
Divide 60 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 7, because the next number, 8, gives 8 × 8 = 64, which is more than 60: any larger factor has already turned up as the partner of a smaller one.
| Try | Divides 60? | Result |
|---|---|---|
| 1 | yes | 1 × 60 |
| 2 | yes | 2 × 30 |
| 3 | yes | 3 × 20 |
| 4 | yes | 4 × 15 |
| 5 | yes | 5 × 12 |
| 6 | yes | 6 × 10 |
| 7 | no | remainder 4 |
The only number that fails is 7. All the others give a pair.
Prime factorisation of 60
Divide by the smallest prime that goes in exactly, and repeat with the answer until you reach 1.
Reading down the left side: 60 = 2 × 2 × 3 × 5, which is written 22 × 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: (2 + 1) × (1 + 1) × (1 + 1) = 12. This works because a factor of 60 can use each prime anywhere from zero times up to its full power.
The 12 factors add up to 168. Leaving out 60 itself, the rest add up to 108, which is more than 60, so 60 is an abundant number. Of its factors, 4 are odd: 1, 3, 5 and 15.
Divisibility rules applied to 60
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 6 + 0 = 6, which is a multiple of 3.
- By 4: yes. The last two digits make 60, and 60 ÷ 4 = 15.
- 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: no. 60 ÷ 8 leaves remainder 4.
- By 9: no. The digits add up to 6, 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 0 and 6. Their difference is 6, which is not 0 or a multiple of 11.
HCF and LCM with 60
60 and 56. 60 = 22 × 3 × 5 and 56 = 23 × 7. The primes they share, each with its smaller power, give the HCF: 4. Every prime with its larger power gives the LCM: 840. By division: 60 = 56 × 1 + 4; 56 = 4 × 14 + 0, so the last divisor, 4, is the HCF. Check: 4 × 840 = 3360 = 60 × 56.
60 and 63. 60 = 22 × 3 × 5 and 63 = 32 × 7. The primes they share, each with its smaller power, give the HCF: 3. Every prime with its larger power gives the LCM: 1260. By division: 63 = 60 × 1 + 3; 60 = 3 × 20 + 0, so the last divisor, 3, is the HCF. Check: 3 × 1260 = 3780 = 60 × 63.
Multiples of 60
Factors divide a number. Multiples are what you get by multiplying it. The first ten multiples of 60 are 60, 120, 180, 240, 300, 360, 420, 480, 540 and 600. A number has a fixed count of factors but endless multiples.
Common questions
What are the factors of 60?
The factors of 60 are 1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30 and 60. There are 12 of them.
What is the prime factorisation of 60?
60 = 2 × 2 × 3 × 5.
Is 60 a perfect square?
No. 7 × 7 = 49 and 8 × 8 = 64, and 60 lies between them.
What is the greatest factor of 60 other than itself?
30. Divide 60 by its smallest prime factor, 2, to get it.