FactorsAbout 12
A dozen. 12 has more factors than any smaller number, which is why eggs, months and inches come in twelves: a dozen can be shared equally among 2, 3, 4 or 6 people.
Factor pairs of 12
A factor pair is two numbers that multiply to give 12: 1 × 12, 2 × 6 and 3 × 4. Each pair is a way of arranging 12 things in equal rows, so 12 tiles can be laid out as a rectangle in 3 different shapes.
How to find the factors of 12 by hand
Divide 12 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 3, because the next number, 4, gives 4 × 4 = 16, which is more than 12: any larger factor has already turned up as the partner of a smaller one.
| Try | Divides 12? | Result |
|---|---|---|
| 1 | yes | 1 × 12 |
| 2 | yes | 2 × 6 |
| 3 | yes | 3 × 4 |
Every number from 1 to 3 divides 12.
Prime factorisation of 12
Divide by the smallest prime that goes in exactly, and repeat with the answer until you reach 1.
Reading down the left side: 12 = 2 × 2 × 3, which is written 22 × 3 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: (2 + 1) × (1 + 1) = 6. This works because a factor of 12 can use each prime anywhere from zero times up to its full power.
The 6 factors add up to 28. Leaving out 12 itself, the rest add up to 16, which is more than 12, so 12 is an abundant number. Of its factors, 2 are odd: 1 and 3.
Divisibility rules applied to 12
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 1 + 2 = 3, which is a multiple of 3.
- By 4: yes. The last two digits make 12, and 12 ÷ 4 = 3.
- 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: no. 12 ÷ 8 leaves remainder 4.
- By 9: no. The digits add up to 3, which is not 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 1. Their difference is 1, which is not 0 or a multiple of 11.
HCF and LCM with 12
12 and 15. 12 = 22 × 3 and 15 = 3 × 5. The primes they share, each with its smaller power, give the HCF: 3. Every prime with its larger power gives the LCM: 60. By division: 15 = 12 × 1 + 3; 12 = 3 × 4 + 0, so the last divisor, 3, is the HCF. Check: 3 × 60 = 180 = 12 × 15.
12 and 15. 12 = 22 × 3 and 15 = 3 × 5. The primes they share, each with its smaller power, give the HCF: 3. Every prime with its larger power gives the LCM: 60. By division: 15 = 12 × 1 + 3; 12 = 3 × 4 + 0, so the last divisor, 3, is the HCF. Check: 3 × 60 = 180 = 12 × 15.
Multiples of 12
Factors divide a number. Multiples are what you get by multiplying it. The first ten multiples of 12 are 12, 24, 36, 48, 60, 72, 84, 96, 108 and 120. A number has a fixed count of factors but endless multiples.
Common questions
What are the factors of 12?
The factors of 12 are 1, 2, 3, 4, 6 and 12. There are 6 of them.
What is the prime factorisation of 12?
12 = 2 × 2 × 3.
Is 12 a perfect square?
No. 3 × 3 = 9 and 4 × 4 = 16, and 12 lies between them.
What is the greatest factor of 12 other than itself?
6. Divide 12 by its smallest prime factor, 2, to get it.