FactorsAbout 225
225 is 15 × 15. Because 15 is 3 × 5, the prime factors of 225 are two 3s and two 5s.
Factor pairs of 225
A factor pair is two numbers that multiply to give 225: 1 × 225, 3 × 75, 5 × 45, 9 × 25 and 15 × 15. Each pair is a way of arranging 225 things in equal rows, so 225 tiles can be laid out as a rectangle in 5 different shapes, one of them a square of side 15.
How to find the factors of 225 by hand
Divide 225 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 15, because 15 × 15 is exactly 225: any larger factor has already turned up as the partner of a smaller one.
| Try | Divides 225? | Result |
|---|---|---|
| 1 | yes | 1 × 225 |
| 2 | no | remainder 1 |
| 3 | yes | 3 × 75 |
| 4 | no | remainder 1 |
| 5 | yes | 5 × 45 |
| 6 | no | remainder 3 |
| 7 | no | remainder 1 |
| 8 | no | remainder 1 |
| 9 | yes | 9 × 25 |
| 10 | no | remainder 5 |
| 11 | no | remainder 5 |
| 12 | no | remainder 9 |
| 13 | no | remainder 4 |
| 14 | no | remainder 1 |
| 15 | yes | 15 × 15 |
The numbers that fail are 2, 4, 6, 7, 8, 10, 11, 12, 13 and 14. All the others give a pair.
Prime factorisation of 225
Divide by the smallest prime that goes in exactly, and repeat with the answer until you reach 1.
Reading down the left side: 225 = 3 × 3 × 5 × 5, which is written 32 × 52 with powers. Its prime factors are 3 and 5.
Counting the factors without listing them
Add 1 to each power in the prime factorisation and multiply the results: (2 + 1) × (2 + 1) = 9. This works because a factor of 225 can use each prime anywhere from zero times up to its full power.
The 9 factors add up to 403. Leaving out 225 itself, the rest add up to 178, which is less than 225, so 225 is a deficient number. Of its factors, 9 are odd: 1, 3, 5, 9, 15, 25, 45, 75 and 225.
Divisibility rules applied to 225
The quick tests agree with the factor list above.
- By 2: no. The last digit is 5, which is odd.
- By 3: yes. The digits add up to 2 + 2 + 5 = 9, which is a multiple of 3.
- By 4: no. The last two digits make 25, and 25 ÷ 4 leaves remainder 1.
- By 5: yes. The last digit is 5.
- By 6: no. A number divides by 6 when it divides by both 2 and 3. Here the test for 2 fails.
- By 8: no. 225 ÷ 8 leaves remainder 1.
- 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 7 and 2. Their difference is 5, which is not 0 or a multiple of 11.
HCF and LCM with 225
225 and 216. 225 = 32 × 52 and 216 = 23 × 33. The primes they share, each with its smaller power, give the HCF: 9. Every prime with its larger power gives the LCM: 5400. By division: 225 = 216 × 1 + 9; 216 = 9 × 24 + 0, so the last divisor, 9, is the HCF. Check: 9 × 5400 = 48600 = 225 × 216.
225 and 240. 225 = 32 × 52 and 240 = 24 × 3 × 5. The primes they share, each with its smaller power, give the HCF: 15. Every prime with its larger power gives the LCM: 3600. By division: 240 = 225 × 1 + 15; 225 = 15 × 15 + 0, so the last divisor, 15, is the HCF. Check: 15 × 3600 = 54000 = 225 × 240.
Multiples of 225
Factors divide a number. Multiples are what you get by multiplying it. The first ten multiples of 225 are 225, 450, 675, 900, 1125, 1350, 1575, 1800, 2025 and 2250. A number has a fixed count of factors but endless multiples.
Common questions
What are the factors of 225?
The factors of 225 are 1, 3, 5, 9, 15, 25, 45, 75 and 225. There are 9 of them.
What is the prime factorisation of 225?
225 = 3 × 3 × 5 × 5.
Is 225 a perfect square?
Yes. 225 = 15 × 15, so its square root is 15.
What is the greatest factor of 225 other than itself?
75. Divide 225 by its smallest prime factor, 3, to get it.