FactorsAbout 125
125 is 5 × 5 × 5, the cube of 5, and one eighth of 1000.
Factor pairs of 125
A factor pair is two numbers that multiply to give 125: 1 × 125 and 5 × 25. Each pair is a way of arranging 125 things in equal rows, so 125 tiles can be laid out as a rectangle in 2 different shapes.
How to find the factors of 125 by hand
Divide 125 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 11, because the next number, 12, gives 12 × 12 = 144, which is more than 125: any larger factor has already turned up as the partner of a smaller one.
| Try | Divides 125? | Result |
|---|---|---|
| 1 | yes | 1 × 125 |
| 2 | no | remainder 1 |
| 3 | no | remainder 2 |
| 4 | no | remainder 1 |
| 5 | yes | 5 × 25 |
| 6 | no | remainder 5 |
| 7 | no | remainder 6 |
| 8 | no | remainder 5 |
| 9 | no | remainder 8 |
| 10 | no | remainder 5 |
| 11 | no | remainder 4 |
The numbers that fail are 2, 3, 4, 6, 7, 8, 9, 10 and 11. All the others give a pair.
Prime factorisation of 125
Divide by the smallest prime that goes in exactly, and repeat with the answer until you reach 1.
Reading down the left side: 125 = 5 × 5 × 5, which is written 53 with powers. It has only one prime factor, 5.
Counting the factors without listing them
Add 1 to each power in the prime factorisation and multiply the results: (3 + 1) = 4. This works because a factor of 125 can use each prime anywhere from zero times up to its full power.
The 4 factors add up to 156. Leaving out 125 itself, the rest add up to 31, which is less than 125, so 125 is a deficient number. Of its factors, 4 are odd: 1, 5, 25 and 125.
Divisibility rules applied to 125
The quick tests agree with the factor list above.
- By 2: no. The last digit is 5, which is odd.
- By 3: no. The digits add up to 1 + 2 + 5 = 8, which is not 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. 125 ÷ 8 leaves remainder 5.
- By 9: no. The digits add up to 8, 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 6 and 2. Their difference is 4, which is not 0 or a multiple of 11.
HCF and LCM with 125
125 and 120. 125 = 53 and 120 = 23 × 3 × 5. The primes they share, each with its smaller power, give the HCF: 5. Every prime with its larger power gives the LCM: 3000. By division: 125 = 120 × 1 + 5; 120 = 5 × 24 + 0, so the last divisor, 5, is the HCF. Check: 5 × 3000 = 15000 = 125 × 120.
125 and 144. 125 = 53 and 144 = 24 × 32. The primes they share, each with its smaller power, give the HCF: 1. Every prime with its larger power gives the LCM: 18000. By division: 144 = 125 × 1 + 19; 125 = 19 × 6 + 11; 19 = 11 × 1 + 8; 11 = 8 × 1 + 3; 8 = 3 × 2 + 2; 3 = 2 × 1 + 1; 2 = 1 × 2 + 0, so the last divisor, 1, is the HCF. Check: 1 × 18000 = 18000 = 125 × 144.
Multiples of 125
Factors divide a number. Multiples are what you get by multiplying it. The first ten multiples of 125 are 125, 250, 375, 500, 625, 750, 875, 1000, 1125 and 1250. A number has a fixed count of factors but endless multiples.
Common questions
What are the factors of 125?
The factors of 125 are 1, 5, 25 and 125. There are 4 of them.
What is the prime factorisation of 125?
125 = 5 × 5 × 5.
Is 125 a perfect square?
No. 11 × 11 = 121 and 12 × 12 = 144, and 125 lies between them.
What is the greatest factor of 125 other than itself?
25. Divide 125 by its smallest prime factor, 5, to get it.