FactorsAbout 90
A right angle is 90 degrees. 90 is 9 × 10, so it passes the divisibility tests for 2, 3, 5, 9 and 10.
Factor pairs of 90
A factor pair is two numbers that multiply to give 90: 1 × 90, 2 × 45, 3 × 30, 5 × 18, 6 × 15 and 9 × 10. Each pair is a way of arranging 90 things in equal rows, so 90 tiles can be laid out as a rectangle in 6 different shapes.
How to find the factors of 90 by hand
Divide 90 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 9, because the next number, 10, gives 10 × 10 = 100, which is more than 90: any larger factor has already turned up as the partner of a smaller one.
| Try | Divides 90? | Result |
|---|---|---|
| 1 | yes | 1 × 90 |
| 2 | yes | 2 × 45 |
| 3 | yes | 3 × 30 |
| 4 | no | remainder 2 |
| 5 | yes | 5 × 18 |
| 6 | yes | 6 × 15 |
| 7 | no | remainder 6 |
| 8 | no | remainder 2 |
| 9 | yes | 9 × 10 |
The numbers that fail are 4, 7 and 8. All the others give a pair.
Prime factorisation of 90
Divide by the smallest prime that goes in exactly, and repeat with the answer until you reach 1.
Reading down the left side: 90 = 2 × 3 × 3 × 5, which is written 2 × 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: (1 + 1) × (2 + 1) × (1 + 1) = 12. This works because a factor of 90 can use each prime anywhere from zero times up to its full power.
The 12 factors add up to 234. Leaving out 90 itself, the rest add up to 144, which is more than 90, so 90 is an abundant number. Of its factors, 6 are odd: 1, 3, 5, 9, 15 and 45.
Divisibility rules applied to 90
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 9 + 0 = 9, which is a multiple of 3.
- By 4: no. The last two digits make 90, and 90 ÷ 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. 90 ÷ 8 leaves remainder 2.
- 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 0 and 9. Their difference is 9, which is not 0 or a multiple of 11.
HCF and LCM with 90
90 and 84. 90 = 2 × 32 × 5 and 84 = 22 × 3 × 7. The primes they share, each with its smaller power, give the HCF: 6. Every prime with its larger power gives the LCM: 1260. By division: 90 = 84 × 1 + 6; 84 = 6 × 14 + 0, so the last divisor, 6, is the HCF. Check: 6 × 1260 = 7560 = 90 × 84.
90 and 96. 90 = 2 × 32 × 5 and 96 = 25 × 3. The primes they share, each with its smaller power, give the HCF: 6. Every prime with its larger power gives the LCM: 1440. By division: 96 = 90 × 1 + 6; 90 = 6 × 15 + 0, so the last divisor, 6, is the HCF. Check: 6 × 1440 = 8640 = 90 × 96.
Multiples of 90
Factors divide a number. Multiples are what you get by multiplying it. The first ten multiples of 90 are 90, 180, 270, 360, 450, 540, 630, 720, 810 and 900. A number has a fixed count of factors but endless multiples.
Common questions
What are the factors of 90?
The factors of 90 are 1, 2, 3, 5, 6, 9, 10, 15, 18, 30, 45 and 90. There are 12 of them.
What is the prime factorisation of 90?
90 = 2 × 3 × 3 × 5.
Is 90 a perfect square?
No. 9 × 9 = 81 and 10 × 10 = 100, and 90 lies between them.
What is the greatest factor of 90 other than itself?
45. Divide 90 by its smallest prime factor, 2, to get it.