FactorsAbout 28
28 is a perfect number: its factors other than itself, 1 + 2 + 4 + 7 + 14, add up to exactly 28. The only smaller perfect number is 6.
Factor pairs of 28
A factor pair is two numbers that multiply to give 28: 1 × 28, 2 × 14 and 4 × 7. Each pair is a way of arranging 28 things in equal rows, so 28 tiles can be laid out as a rectangle in 3 different shapes.
How to find the factors of 28 by hand
Divide 28 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 5, because the next number, 6, gives 6 × 6 = 36, which is more than 28: any larger factor has already turned up as the partner of a smaller one.
| Try | Divides 28? | Result |
|---|---|---|
| 1 | yes | 1 × 28 |
| 2 | yes | 2 × 14 |
| 3 | no | remainder 1 |
| 4 | yes | 4 × 7 |
| 5 | no | remainder 3 |
The numbers that fail are 3 and 5. All the others give a pair.
Prime factorisation of 28
Divide by the smallest prime that goes in exactly, and repeat with the answer until you reach 1.
Reading down the left side: 28 = 2 × 2 × 7, which is written 22 × 7 with powers. Its prime factors are 2 and 7.
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 28 can use each prime anywhere from zero times up to its full power.
The 6 factors add up to 56. Leaving out 28 itself, the rest add up to 28, which is no more and no less than 28, so 28 is a perfect number. Of its factors, 2 are odd: 1 and 7.
Divisibility rules applied to 28
The quick tests agree with the factor list above.
- By 2: yes. The last digit is 8, which is even.
- By 3: no. The digits add up to 2 + 8 = 10, which is not a multiple of 3.
- By 4: yes. The last two digits make 28, and 28 ÷ 4 = 7.
- By 5: no. The last digit is 8, not 0 or 5.
- By 6: no. A number divides by 6 when it divides by both 2 and 3. Here the test for 3 fails.
- By 8: no. 28 ÷ 8 leaves remainder 4.
- By 9: no. The digits add up to 10, 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 2. Their difference is 6, which is not 0 or a multiple of 11.
HCF and LCM with 28
28 and 25. 28 = 22 × 7 and 25 = 52. The primes they share, each with its smaller power, give the HCF: 1. Every prime with its larger power gives the LCM: 700. By division: 28 = 25 × 1 + 3; 25 = 3 × 8 + 1; 3 = 1 × 3 + 0, so the last divisor, 1, is the HCF. Check: 1 × 700 = 700 = 28 × 25.
28 and 30. 28 = 22 × 7 and 30 = 2 × 3 × 5. The primes they share, each with its smaller power, give the HCF: 2. Every prime with its larger power gives the LCM: 420. By division: 30 = 28 × 1 + 2; 28 = 2 × 14 + 0, so the last divisor, 2, is the HCF. Check: 2 × 420 = 840 = 28 × 30.
Multiples of 28
Factors divide a number. Multiples are what you get by multiplying it. The first ten multiples of 28 are 28, 56, 84, 112, 140, 168, 196, 224, 252 and 280. A number has a fixed count of factors but endless multiples.
Common questions
What are the factors of 28?
The factors of 28 are 1, 2, 4, 7, 14 and 28. There are 6 of them.
What is the prime factorisation of 28?
28 = 2 × 2 × 7.
Is 28 a perfect square?
No. 5 × 5 = 25 and 6 × 6 = 36, and 28 lies between them.
What is the greatest factor of 28 other than itself?
14. Divide 28 by its smallest prime factor, 2, to get it.