FactorsAbout 180
The angles of a triangle add up to 180 degrees, and a straight line is an angle of 180 degrees. 180 has eighteen factors, more than any smaller number.
Factor pairs of 180
A factor pair is two numbers that multiply to give 180: 1 × 180, 2 × 90, 3 × 60, 4 × 45, 5 × 36, 6 × 30, 9 × 20, 10 × 18 and 12 × 15. Each pair is a way of arranging 180 things in equal rows, so 180 tiles can be laid out as a rectangle in 9 different shapes.
How to find the factors of 180 by hand
Divide 180 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 13, because the next number, 14, gives 14 × 14 = 196, which is more than 180: any larger factor has already turned up as the partner of a smaller one.
| Try | Divides 180? | Result |
|---|---|---|
| 1 | yes | 1 × 180 |
| 2 | yes | 2 × 90 |
| 3 | yes | 3 × 60 |
| 4 | yes | 4 × 45 |
| 5 | yes | 5 × 36 |
| 6 | yes | 6 × 30 |
| 7 | no | remainder 5 |
| 8 | no | remainder 4 |
| 9 | yes | 9 × 20 |
| 10 | yes | 10 × 18 |
| 11 | no | remainder 4 |
| 12 | yes | 12 × 15 |
| 13 | no | remainder 11 |
The numbers that fail are 7, 8, 11 and 13. All the others give a pair.
Prime factorisation of 180
Divide by the smallest prime that goes in exactly, and repeat with the answer until you reach 1.
Reading down the left side: 180 = 2 × 2 × 3 × 3 × 5, which is written 22 × 32 × 5 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: (2 + 1) × (2 + 1) × (1 + 1) = 18. This works because a factor of 180 can use each prime anywhere from zero times up to its full power.
The 18 factors add up to 546. Leaving out 180 itself, the rest add up to 366, which is more than 180, so 180 is an abundant number. Of its factors, 6 are odd: 1, 3, 5, 9, 15 and 45.
Divisibility rules applied to 180
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 + 8 + 0 = 9, which is a multiple of 3.
- By 4: yes. The last two digits make 80, and 80 ÷ 4 = 20.
- 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. 180 ÷ 8 leaves remainder 4.
- By 9: yes. The digits add up to 9, which is a multiple of 9.
- By 10: yes. The last digit is 0.
- By 11: no. Adding alternate digits from the right gives 1 and 8. Their difference is 7, which is not 0 or a multiple of 11.
HCF and LCM with 180
180 and 150. 180 = 22 × 32 × 5 and 150 = 2 × 3 × 52. 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 = 180 × 150.
180 and 200. 180 = 22 × 32 × 5 and 200 = 23 × 52. The primes they share, each with its smaller power, give the HCF: 20. Every prime with its larger power gives the LCM: 1800. By division: 200 = 180 × 1 + 20; 180 = 20 × 9 + 0, so the last divisor, 20, is the HCF. Check: 20 × 1800 = 36000 = 180 × 200.
Multiples of 180
Factors divide a number. Multiples are what you get by multiplying it. The first ten multiples of 180 are 180, 360, 540, 720, 900, 1080, 1260, 1440, 1620 and 1800. A number has a fixed count of factors but endless multiples.
Common questions
What are the factors of 180?
The factors of 180 are 1, 2, 3, 4, 5, 6, 9, 10, 12, 15, 18, 20, 30, 36, 45, 60, 90 and 180. There are 18 of them.
What is the prime factorisation of 180?
180 = 2 × 2 × 3 × 3 × 5.
Is 180 a perfect square?
No. 13 × 13 = 169 and 14 × 14 = 196, and 180 lies between them.
What is the greatest factor of 180 other than itself?
90. Divide 180 by its smallest prime factor, 2, to get it.