FactorsAbout 48
48 has ten factors, more than any smaller number. Four dozen is 48.
Factor pairs of 48
A factor pair is two numbers that multiply to give 48: 1 × 48, 2 × 24, 3 × 16, 4 × 12 and 6 × 8. Each pair is a way of arranging 48 things in equal rows, so 48 tiles can be laid out as a rectangle in 5 different shapes.
How to find the factors of 48 by hand
Divide 48 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 6, because the next number, 7, gives 7 × 7 = 49, which is more than 48: any larger factor has already turned up as the partner of a smaller one.
| Try | Divides 48? | Result |
|---|---|---|
| 1 | yes | 1 × 48 |
| 2 | yes | 2 × 24 |
| 3 | yes | 3 × 16 |
| 4 | yes | 4 × 12 |
| 5 | no | remainder 3 |
| 6 | yes | 6 × 8 |
The only number that fails is 5. All the others give a pair.
Prime factorisation of 48
Divide by the smallest prime that goes in exactly, and repeat with the answer until you reach 1.
Reading down the left side: 48 = 2 × 2 × 2 × 2 × 3, which is written 24 × 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: (4 + 1) × (1 + 1) = 10. This works because a factor of 48 can use each prime anywhere from zero times up to its full power.
The 10 factors add up to 124. Leaving out 48 itself, the rest add up to 76, which is more than 48, so 48 is an abundant number. Of its factors, 2 are odd: 1 and 3.
Divisibility rules applied to 48
The quick tests agree with the factor list above.
- By 2: yes. The last digit is 8, which is even.
- By 3: yes. The digits add up to 4 + 8 = 12, which is a multiple of 3.
- By 4: yes. The last two digits make 48, and 48 ÷ 4 = 12.
- By 5: no. The last digit is 8, 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. 48 ÷ 8 = 6.
- By 9: no. The digits add up to 12, 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 8 and 4. Their difference is 4, which is not 0 or a multiple of 11.
HCF and LCM with 48
48 and 45. 48 = 24 × 3 and 45 = 32 × 5. The primes they share, each with its smaller power, give the HCF: 3. Every prime with its larger power gives the LCM: 720. By division: 48 = 45 × 1 + 3; 45 = 3 × 15 + 0, so the last divisor, 3, is the HCF. Check: 3 × 720 = 2160 = 48 × 45.
48 and 50. 48 = 24 × 3 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: 1200. By division: 50 = 48 × 1 + 2; 48 = 2 × 24 + 0, so the last divisor, 2, is the HCF. Check: 2 × 1200 = 2400 = 48 × 50.
Multiples of 48
Factors divide a number. Multiples are what you get by multiplying it. The first ten multiples of 48 are 48, 96, 144, 192, 240, 288, 336, 384, 432 and 480. A number has a fixed count of factors but endless multiples.
Common questions
What are the factors of 48?
The factors of 48 are 1, 2, 3, 4, 6, 8, 12, 16, 24 and 48. There are 10 of them.
What is the prime factorisation of 48?
48 = 2 × 2 × 2 × 2 × 3.
Is 48 a perfect square?
No. 6 × 6 = 36 and 7 × 7 = 49, and 48 lies between them.
What is the greatest factor of 48 other than itself?
24. Divide 48 by its smallest prime factor, 2, to get it.