FactorsAbout 72
72 is 8 × 9 and has twelve factors. Six dozen is 72.
Factor pairs of 72
A factor pair is two numbers that multiply to give 72: 1 × 72, 2 × 36, 3 × 24, 4 × 18, 6 × 12 and 8 × 9. Each pair is a way of arranging 72 things in equal rows, so 72 tiles can be laid out as a rectangle in 6 different shapes.
How to find the factors of 72 by hand
Divide 72 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 8, because the next number, 9, gives 9 × 9 = 81, which is more than 72: any larger factor has already turned up as the partner of a smaller one.
| Try | Divides 72? | Result |
|---|---|---|
| 1 | yes | 1 × 72 |
| 2 | yes | 2 × 36 |
| 3 | yes | 3 × 24 |
| 4 | yes | 4 × 18 |
| 5 | no | remainder 2 |
| 6 | yes | 6 × 12 |
| 7 | no | remainder 2 |
| 8 | yes | 8 × 9 |
The numbers that fail are 5 and 7. All the others give a pair.
Prime factorisation of 72
Divide by the smallest prime that goes in exactly, and repeat with the answer until you reach 1.
Reading down the left side: 72 = 2 × 2 × 2 × 3 × 3, which is written 23 × 32 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) × (2 + 1) = 12. This works because a factor of 72 can use each prime anywhere from zero times up to its full power.
The 12 factors add up to 195. Leaving out 72 itself, the rest add up to 123, which is more than 72, so 72 is an abundant number. Of its factors, 3 are odd: 1, 3 and 9.
Divisibility rules applied to 72
The quick tests agree with the factor list above.
- By 2: yes. The last digit is 2, which is even.
- By 3: yes. The digits add up to 7 + 2 = 9, which is a multiple of 3.
- By 4: yes. The last two digits make 72, and 72 ÷ 4 = 18.
- By 5: no. The last digit is 2, 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. 72 ÷ 8 = 9.
- 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 2 and 7. Their difference is 5, which is not 0 or a multiple of 11.
HCF and LCM with 72
72 and 64. 72 = 23 × 32 and 64 = 26. The primes they share, each with its smaller power, give the HCF: 8. Every prime with its larger power gives the LCM: 576. By division: 72 = 64 × 1 + 8; 64 = 8 × 8 + 0, so the last divisor, 8, is the HCF. Check: 8 × 576 = 4608 = 72 × 64.
72 and 75. 72 = 23 × 32 and 75 = 3 × 52. The primes they share, each with its smaller power, give the HCF: 3. Every prime with its larger power gives the LCM: 1800. By division: 75 = 72 × 1 + 3; 72 = 3 × 24 + 0, so the last divisor, 3, is the HCF. Check: 3 × 1800 = 5400 = 72 × 75.
Multiples of 72
Factors divide a number. Multiples are what you get by multiplying it. The first ten multiples of 72 are 72, 144, 216, 288, 360, 432, 504, 576, 648 and 720. A number has a fixed count of factors but endless multiples.
Common questions
What are the factors of 72?
The factors of 72 are 1, 2, 3, 4, 6, 8, 9, 12, 18, 24, 36 and 72. There are 12 of them.
What is the prime factorisation of 72?
72 = 2 × 2 × 2 × 3 × 3.
Is 72 a perfect square?
No. 8 × 8 = 64 and 9 × 9 = 81, and 72 lies between them.
What is the greatest factor of 72 other than itself?
36. Divide 72 by its smallest prime factor, 2, to get it.