FactorsAbout 216
216 is 6 × 6 × 6, the cube of 6.
Factor pairs of 216
A factor pair is two numbers that multiply to give 216: 1 × 216, 2 × 108, 3 × 72, 4 × 54, 6 × 36, 8 × 27, 9 × 24 and 12 × 18. Each pair is a way of arranging 216 things in equal rows, so 216 tiles can be laid out as a rectangle in 8 different shapes.
How to find the factors of 216 by hand
Divide 216 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 14, because the next number, 15, gives 15 × 15 = 225, which is more than 216: any larger factor has already turned up as the partner of a smaller one.
| Try | Divides 216? | Result |
|---|---|---|
| 1 | yes | 1 × 216 |
| 2 | yes | 2 × 108 |
| 3 | yes | 3 × 72 |
| 4 | yes | 4 × 54 |
| 5 | no | remainder 1 |
| 6 | yes | 6 × 36 |
| 7 | no | remainder 6 |
| 8 | yes | 8 × 27 |
| 9 | yes | 9 × 24 |
| 10 | no | remainder 6 |
| 11 | no | remainder 7 |
| 12 | yes | 12 × 18 |
| 13 | no | remainder 8 |
| 14 | no | remainder 6 |
The numbers that fail are 5, 7, 10, 11, 13 and 14. All the others give a pair.
Prime factorisation of 216
Divide by the smallest prime that goes in exactly, and repeat with the answer until you reach 1.
Reading down the left side: 216 = 2 × 2 × 2 × 3 × 3 × 3, which is written 23 × 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: (3 + 1) × (3 + 1) = 16. This works because a factor of 216 can use each prime anywhere from zero times up to its full power.
The 16 factors add up to 600. Leaving out 216 itself, the rest add up to 384, which is more than 216, so 216 is an abundant number. Of its factors, 4 are odd: 1, 3, 9 and 27.
Divisibility rules applied to 216
The quick tests agree with the factor list above.
- By 2: yes. The last digit is 6, which is even.
- By 3: yes. The digits add up to 2 + 1 + 6 = 9, which is a multiple of 3.
- By 4: yes. The last two digits make 16, and 16 ÷ 4 = 4.
- By 5: no. The last digit is 6, 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: yes. 216 ÷ 8 = 27.
- 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 8 and 1. Their difference is 7, which is not 0 or a multiple of 11.
HCF and LCM with 216
216 and 200. 216 = 23 × 33 and 200 = 23 × 52. The primes they share, each with its smaller power, give the HCF: 8. Every prime with its larger power gives the LCM: 5400. By division: 216 = 200 × 1 + 16; 200 = 16 × 12 + 8; 16 = 8 × 2 + 0, so the last divisor, 8, is the HCF. Check: 8 × 5400 = 43200 = 216 × 200.
216 and 225. 216 = 23 × 33 and 225 = 32 × 52. The primes they share, each with its smaller power, give the HCF: 9. Every prime with its larger power gives the LCM: 5400. By division: 225 = 216 × 1 + 9; 216 = 9 × 24 + 0, so the last divisor, 9, is the HCF. Check: 9 × 5400 = 48600 = 216 × 225.
Multiples of 216
Factors divide a number. Multiples are what you get by multiplying it. The first ten multiples of 216 are 216, 432, 648, 864, 1080, 1296, 1512, 1728, 1944 and 2160. A number has a fixed count of factors but endless multiples.
Common questions
What are the factors of 216?
The factors of 216 are 1, 2, 3, 4, 6, 8, 9, 12, 18, 24, 27, 36, 54, 72, 108 and 216. There are 16 of them.
What is the prime factorisation of 216?
216 = 2 × 2 × 2 × 3 × 3 × 3.
Is 216 a perfect square?
No. 14 × 14 = 196 and 15 × 15 = 225, and 216 lies between them.
What is the greatest factor of 216 other than itself?
108. Divide 216 by its smallest prime factor, 2, to get it.