FactorsAbout 1000
A thousand is 10 × 10 × 10. A kilometre is 1000 metres and a kilogram is 1000 grams.
Factor pairs of 1000
A factor pair is two numbers that multiply to give 1000: 1 × 1000, 2 × 500, 4 × 250, 5 × 200, 8 × 125, 10 × 100, 20 × 50 and 25 × 40. Each pair is a way of arranging 1000 things in equal rows, so 1000 tiles can be laid out as a rectangle in 8 different shapes.
How to find the factors of 1000 by hand
Divide 1000 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 31, because the next number, 32, gives 32 × 32 = 1024, which is more than 1000: any larger factor has already turned up as the partner of a smaller one.
| Try | Divides 1000? | Result |
|---|---|---|
| 1 | yes | 1 × 1000 |
| 2 | yes | 2 × 500 |
| 3 | no | remainder 1 |
| 4 | yes | 4 × 250 |
| 5 | yes | 5 × 200 |
| 6 | no | remainder 4 |
| 7 | no | remainder 6 |
| 8 | yes | 8 × 125 |
| 9 | no | remainder 1 |
| 10 | yes | 10 × 100 |
| 11 | no | remainder 10 |
| 12 | no | remainder 4 |
| 13 | no | remainder 12 |
| 14 | no | remainder 6 |
| 15 | no | remainder 10 |
| 16 | no | remainder 8 |
| 17 | no | remainder 14 |
| 18 | no | remainder 10 |
| 19 | no | remainder 12 |
| 20 | yes | 20 × 50 |
| 21 | no | remainder 13 |
| 22 | no | remainder 10 |
| 23 | no | remainder 11 |
| 24 | no | remainder 16 |
| 25 | yes | 25 × 40 |
| 26 | no | remainder 12 |
| 27 | no | remainder 1 |
| 28 | no | remainder 20 |
| 29 | no | remainder 14 |
| 30 | no | remainder 10 |
| 31 | no | remainder 8 |
The numbers that fail are 3, 6, 7, 9, 11, 12, 13, 14, 15, 16, 17, 18, 19, 21, 22, 23, 24, 26, 27, 28, 29, 30 and 31. All the others give a pair.
Prime factorisation of 1000
Divide by the smallest prime that goes in exactly, and repeat with the answer until you reach 1.
Reading down the left side: 1000 = 2 × 2 × 2 × 5 × 5 × 5, which is written 23 × 53 with powers. Its prime factors are 2 and 5.
Counting the factors without listing them
Add 1 to each power in the prime factorisation and multiply the results: (3 + 1) × (3 + 1) = 16. This works because a factor of 1000 can use each prime anywhere from zero times up to its full power.
The 16 factors add up to 2340. Leaving out 1000 itself, the rest add up to 1340, which is more than 1000, so 1000 is an abundant number. Of its factors, 4 are odd: 1, 5, 25 and 125.
Divisibility rules applied to 1000
The quick tests agree with the factor list above.
- By 2: yes. The last digit is 0, which is even.
- By 3: no. The digits add up to 1 + 0 + 0 + 0 = 1, which is not a multiple of 3.
- By 4: yes. The last two digits make 0, and 0 ÷ 4 = 0.
- By 5: yes. The last digit is 0.
- By 6: no. A number divides by 6 when it divides by both 2 and 3. Here the test for 3 fails.
- By 8: yes. The last three digits make 0, and 0 ÷ 8 = 0.
- By 9: no. The digits add up to 1, which is not a multiple of 9.
- By 10: yes. The last digit is 0.
- By 11: no. Adding alternate digits from the right gives 0 and 1. Their difference is 1, which is not 0 or a multiple of 11.
HCF and LCM with 1000
1000 and 720. 1000 = 23 × 53 and 720 = 24 × 32 × 5. The primes they share, each with its smaller power, give the HCF: 40. Every prime with its larger power gives the LCM: 18000. By division: 1000 = 720 × 1 + 280; 720 = 280 × 2 + 160; 280 = 160 × 1 + 120; 160 = 120 × 1 + 40; 120 = 40 × 3 + 0, so the last divisor, 40, is the HCF. Check: 40 × 18000 = 720000 = 1000 × 720.
1000 and 625. 1000 = 23 × 53 and 625 = 54. The primes they share, each with its smaller power, give the HCF: 125. Every prime with its larger power gives the LCM: 5000. By division: 1000 = 625 × 1 + 375; 625 = 375 × 1 + 250; 375 = 250 × 1 + 125; 250 = 125 × 2 + 0, so the last divisor, 125, is the HCF. Check: 125 × 5000 = 625000 = 1000 × 625.
Multiples of 1000
Factors divide a number. Multiples are what you get by multiplying it. The first ten multiples of 1000 are 1000, 2000, 3000, 4000, 5000, 6000, 7000, 8000, 9000 and 10000. A number has a fixed count of factors but endless multiples.
Common questions
What are the factors of 1000?
The factors of 1000 are 1, 2, 4, 5, 8, 10, 20, 25, 40, 50, 100, 125, 200, 250, 500 and 1000. There are 16 of them.
What is the prime factorisation of 1000?
1000 = 2 × 2 × 2 × 5 × 5 × 5.
Is 1000 a perfect square?
No. 31 × 31 = 961 and 32 × 32 = 1024, and 1000 lies between them.
What is the greatest factor of 1000 other than itself?
500. Divide 1000 by its smallest prime factor, 2, to get it.