FactorsAbout 360
A full turn is 360 degrees. 360 has twenty-four factors, more than any smaller number, so a circle can be cut into equal slices in many different ways.
Factor pairs of 360
A factor pair is two numbers that multiply to give 360: 1 × 360, 2 × 180, 3 × 120, 4 × 90, 5 × 72, 6 × 60, 8 × 45, 9 × 40, 10 × 36, 12 × 30, 15 × 24 and 18 × 20. Each pair is a way of arranging 360 things in equal rows, so 360 tiles can be laid out as a rectangle in 12 different shapes.
How to find the factors of 360 by hand
Divide 360 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 18, because the next number, 19, gives 19 × 19 = 361, which is more than 360: any larger factor has already turned up as the partner of a smaller one.
| Try | Divides 360? | Result |
|---|---|---|
| 1 | yes | 1 × 360 |
| 2 | yes | 2 × 180 |
| 3 | yes | 3 × 120 |
| 4 | yes | 4 × 90 |
| 5 | yes | 5 × 72 |
| 6 | yes | 6 × 60 |
| 7 | no | remainder 3 |
| 8 | yes | 8 × 45 |
| 9 | yes | 9 × 40 |
| 10 | yes | 10 × 36 |
| 11 | no | remainder 8 |
| 12 | yes | 12 × 30 |
| 13 | no | remainder 9 |
| 14 | no | remainder 10 |
| 15 | yes | 15 × 24 |
| 16 | no | remainder 8 |
| 17 | no | remainder 3 |
| 18 | yes | 18 × 20 |
The numbers that fail are 7, 11, 13, 14, 16 and 17. All the others give a pair.
Prime factorisation of 360
Divide by the smallest prime that goes in exactly, and repeat with the answer until you reach 1.
Reading down the left side: 360 = 2 × 2 × 2 × 3 × 3 × 5, which is written 23 × 32 × 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) × (2 + 1) × (1 + 1) = 24. This works because a factor of 360 can use each prime anywhere from zero times up to its full power.
The 24 factors add up to 1170. Leaving out 360 itself, the rest add up to 810, which is more than 360, so 360 is an abundant number. Of its factors, 6 are odd: 1, 3, 5, 9, 15 and 45.
Divisibility rules applied to 360
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 3 + 6 + 0 = 9, 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: yes. 360 ÷ 8 = 45.
- By 9: yes. The digits add up to 9, which is a multiple of 9.
- By 10: yes. The last digit is 0.
- By 11: no. Adding alternate digits from the right gives 3 and 6. Their difference is 3, which is not 0 or a multiple of 11.
HCF and LCM with 360
360 and 300. 360 = 23 × 32 × 5 and 300 = 22 × 3 × 52. The primes they share, each with its smaller power, give the HCF: 60. Every prime with its larger power gives the LCM: 1800. By division: 360 = 300 × 1 + 60; 300 = 60 × 5 + 0, so the last divisor, 60, is the HCF. Check: 60 × 1800 = 108000 = 360 × 300.
360 and 400. 360 = 23 × 32 × 5 and 400 = 24 × 52. The primes they share, each with its smaller power, give the HCF: 40. Every prime with its larger power gives the LCM: 3600. By division: 400 = 360 × 1 + 40; 360 = 40 × 9 + 0, so the last divisor, 40, is the HCF. Check: 40 × 3600 = 144000 = 360 × 400.
Multiples of 360
Factors divide a number. Multiples are what you get by multiplying it. The first ten multiples of 360 are 360, 720, 1080, 1440, 1800, 2160, 2520, 2880, 3240 and 3600. A number has a fixed count of factors but endless multiples.
Common questions
What are the factors of 360?
The factors of 360 are 1, 2, 3, 4, 5, 6, 8, 9, 10, 12, 15, 18, 20, 24, 30, 36, 40, 45, 60, 72, 90, 120, 180 and 360. There are 24 of them.
What is the prime factorisation of 360?
360 = 2 × 2 × 2 × 3 × 3 × 5.
Is 360 a perfect square?
No. 18 × 18 = 324 and 19 × 19 = 361, and 360 lies between them.
What is the greatest factor of 360 other than itself?
180. Divide 360 by its smallest prime factor, 2, to get it.