FactorsAbout 150
150 is one and a half hundreds, or 6 × 25. Two and a half hours contain 150 minutes.
Factor pairs of 150
A factor pair is two numbers that multiply to give 150: 1 × 150, 2 × 75, 3 × 50, 5 × 30, 6 × 25 and 10 × 15. Each pair is a way of arranging 150 things in equal rows, so 150 tiles can be laid out as a rectangle in 6 different shapes.
How to find the factors of 150 by hand
Divide 150 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 12, because the next number, 13, gives 13 × 13 = 169, which is more than 150: any larger factor has already turned up as the partner of a smaller one.
| Try | Divides 150? | Result |
|---|---|---|
| 1 | yes | 1 × 150 |
| 2 | yes | 2 × 75 |
| 3 | yes | 3 × 50 |
| 4 | no | remainder 2 |
| 5 | yes | 5 × 30 |
| 6 | yes | 6 × 25 |
| 7 | no | remainder 3 |
| 8 | no | remainder 6 |
| 9 | no | remainder 6 |
| 10 | yes | 10 × 15 |
| 11 | no | remainder 7 |
| 12 | no | remainder 6 |
The numbers that fail are 4, 7, 8, 9, 11 and 12. All the others give a pair.
Prime factorisation of 150
Divide by the smallest prime that goes in exactly, and repeat with the answer until you reach 1.
Reading down the left side: 150 = 2 × 3 × 5 × 5, which is written 2 × 3 × 52 with powers. Its prime factors are 2, 3 and 5.
Counting the factors without listing them
Add 1 to each power in the prime factorisation and multiply the results: (1 + 1) × (1 + 1) × (2 + 1) = 12. This works because a factor of 150 can use each prime anywhere from zero times up to its full power.
The 12 factors add up to 372. Leaving out 150 itself, the rest add up to 222, which is more than 150, so 150 is an abundant number. Of its factors, 6 are odd: 1, 3, 5, 15, 25 and 75.
Divisibility rules applied to 150
The quick tests agree with the factor list above.
- By 2: yes. The last digit is 0, which is even.
- By 3: yes. The digits add up to 1 + 5 + 0 = 6, which is a multiple of 3.
- By 4: no. The last two digits make 50, and 50 ÷ 4 leaves remainder 2.
- By 5: yes. The last digit is 0.
- By 6: yes. A number divides by 6 when it divides by both 2 and 3. Here both tests pass.
- By 8: no. 150 ÷ 8 leaves remainder 6.
- By 9: no. The digits add up to 6, 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 1 and 5. Their difference is 4, which is not 0 or a multiple of 11.
HCF and LCM with 150
150 and 144. 150 = 2 × 3 × 52 and 144 = 24 × 32. The primes they share, each with its smaller power, give the HCF: 6. Every prime with its larger power gives the LCM: 3600. By division: 150 = 144 × 1 + 6; 144 = 6 × 24 + 0, so the last divisor, 6, is the HCF. Check: 6 × 3600 = 21600 = 150 × 144.
150 and 180. 150 = 2 × 3 × 52 and 180 = 22 × 32 × 5. The primes they share, each with its smaller power, give the HCF: 30. Every prime with its larger power gives the LCM: 900. By division: 180 = 150 × 1 + 30; 150 = 30 × 5 + 0, so the last divisor, 30, is the HCF. Check: 30 × 900 = 27000 = 150 × 180.
Multiples of 150
Factors divide a number. Multiples are what you get by multiplying it. The first ten multiples of 150 are 150, 300, 450, 600, 750, 900, 1050, 1200, 1350 and 1500. A number has a fixed count of factors but endless multiples.
Common questions
What are the factors of 150?
The factors of 150 are 1, 2, 3, 5, 6, 10, 15, 25, 30, 50, 75 and 150. There are 12 of them.
What is the prime factorisation of 150?
150 = 2 × 3 × 5 × 5.
Is 150 a perfect square?
No. 12 × 12 = 144 and 13 × 13 = 169, and 150 lies between them.
What is the greatest factor of 150 other than itself?
75. Divide 150 by its smallest prime factor, 2, to get it.