FactorsAbout 54
54 is 2 × 27, twice a cube. In the times tables it is often mixed up with 56: 54 is 6 × 9 and 56 is 7 × 8.
Factor pairs of 54
A factor pair is two numbers that multiply to give 54: 1 × 54, 2 × 27, 3 × 18 and 6 × 9. Each pair is a way of arranging 54 things in equal rows, so 54 tiles can be laid out as a rectangle in 4 different shapes.
How to find the factors of 54 by hand
Divide 54 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 54: any larger factor has already turned up as the partner of a smaller one.
| Try | Divides 54? | Result |
|---|---|---|
| 1 | yes | 1 × 54 |
| 2 | yes | 2 × 27 |
| 3 | yes | 3 × 18 |
| 4 | no | remainder 2 |
| 5 | no | remainder 4 |
| 6 | yes | 6 × 9 |
| 7 | no | remainder 5 |
The numbers that fail are 4, 5 and 7. All the others give a pair.
Prime factorisation of 54
Divide by the smallest prime that goes in exactly, and repeat with the answer until you reach 1.
Reading down the left side: 54 = 2 × 3 × 3 × 3, which is written 2 × 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: (1 + 1) × (3 + 1) = 8. This works because a factor of 54 can use each prime anywhere from zero times up to its full power.
The 8 factors add up to 120. Leaving out 54 itself, the rest add up to 66, which is more than 54, so 54 is an abundant number. Of its factors, 4 are odd: 1, 3, 9 and 27.
Divisibility rules applied to 54
The quick tests agree with the factor list above.
- By 2: yes. The last digit is 4, which is even.
- By 3: yes. The digits add up to 5 + 4 = 9, which is a multiple of 3.
- By 4: no. The last two digits make 54, and 54 ÷ 4 leaves remainder 2.
- By 5: no. The last digit is 4, 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. 54 ÷ 8 leaves remainder 6.
- 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 4 and 5. Their difference is 1, which is not 0 or a multiple of 11.
HCF and LCM with 54
54 and 50. 54 = 2 × 33 and 50 = 2 × 52. The primes they share, each with its smaller power, give the HCF: 2. Every prime with its larger power gives the LCM: 1350. By division: 54 = 50 × 1 + 4; 50 = 4 × 12 + 2; 4 = 2 × 2 + 0, so the last divisor, 2, is the HCF. Check: 2 × 1350 = 2700 = 54 × 50.
54 and 56. 54 = 2 × 33 and 56 = 23 × 7. The primes they share, each with its smaller power, give the HCF: 2. Every prime with its larger power gives the LCM: 1512. By division: 56 = 54 × 1 + 2; 54 = 2 × 27 + 0, so the last divisor, 2, is the HCF. Check: 2 × 1512 = 3024 = 54 × 56.
Multiples of 54
Factors divide a number. Multiples are what you get by multiplying it. The first ten multiples of 54 are 54, 108, 162, 216, 270, 324, 378, 432, 486 and 540. A number has a fixed count of factors but endless multiples.
Common questions
What are the factors of 54?
The factors of 54 are 1, 2, 3, 6, 9, 18, 27 and 54. There are 8 of them.
What is the prime factorisation of 54?
54 = 2 × 3 × 3 × 3.
Is 54 a perfect square?
No. 7 × 7 = 49 and 8 × 8 = 64, and 54 lies between them.
What is the greatest factor of 54 other than itself?
27. Divide 54 by its smallest prime factor, 2, to get it.