Factors of 90
90 has 12 factors: 1, 2, 3, 5, 6, 9, 10, 15, 18, 30, 45 and 90. This page shows the pairs that multiply to make 90, its factor tree, the divisibility rules it passes, its multiples and its square root, with the working for each one.
All 12 factors of 90
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12 factors
Strictly, every factor also has a negative twin, because two negative numbers multiply to a positive one. School questions almost always mean the positive factors, which is what this page lists.
- Factor pairs
- 1 × 90, 2 × 45, 3 × 30, 5 × 18, 6 × 15 and 9 × 10
- Prime factorization
- 90 = 2 × 3² × 5
Why these are the factors of 90
Two numbers form a factor pair of 90 when 90 squares can be arranged into a rectangle with those side lengths and none left over. 90 makes 6 such rectangles, and the factor list is simply their side lengths written down. Tap a pair to build it.
The factor tree of 90
Split 90 into any two factors, then keep splitting until only primes are left. The bottom row gives 90 = 2 × 3 × 3 × 5. Try a different first split: the tree changes shape, but it always ends on the same primes. Click any circled number to split it another way.
90 = 2 × 3 × 3 × 5
However you split it, the bottom row stays the same. Every number has exactly one prime factorization.
Divisibility rules, checked against 90
Most factors of 90 can be found without dividing. Each rule below looks at one small part of the number, such as the last digit or the digit sum, and each row shows the exact value the rule checks, so you can verify it yourself.
| Rule | It looks at | Verdict |
|---|---|---|
| 2 | The last digit is even. Here it is 0. | Yes: 90 ÷ 2 = 45 |
| 3 | The digits add up to a multiple of 3. Here they add to 9. | Yes: 90 ÷ 3 = 30 |
| 4 | The last two digits form a multiple of 4. Here they are 90. | No: 90 ÷ 4 leaves remainder 2 |
| 5 | The last digit is 0 or 5. Here it is 0. | Yes: 90 ÷ 5 = 18 |
| 6 | It passes both the 2 rule and the 3 rule. | Yes: 90 ÷ 6 = 15 |
| 7 | Double the last digit and subtract it from the rest. Here that gives 9. | No: 90 ÷ 7 leaves remainder 6 |
| 8 | The last three digits form a multiple of 8. Here they are 90. | No: 90 ÷ 8 leaves remainder 2 |
| 9 | The digits add up to a multiple of 9. Here they add to 9. | Yes: 90 ÷ 9 = 10 |
| 10 | The last digit is 0. Here it is 0. | Yes: 90 ÷ 10 = 9 |
| 11 | Add and subtract the digits alternately from the right. Here that gives -9. | No: 90 ÷ 11 leaves remainder 2 |
Multiples of 90
Multiples run in the other direction: they are the numbers 90 divides into exactly. The list starts 90, 180, 270, 360, 450, 540 and goes on forever, growing by 90 each step. Every multiple of 90 has 90 in its own factor list.
The square root of 90
√90 is not a whole number, because 90 sits between the perfect squares 81 and 100. Its exact simplified form is 3√10: the square factor moves out from under the root sign. The decimal 9.486833 is a rounded approximation, not the exact value.
Everything else about 90
| Even or odd | Even |
|---|---|
| Prime or composite | Composite |
| Number of factors | 12, which matches the rule from 2 × 3² × 5: add one to each exponent and multiply. |
| Factors added together | 234 |
| Compared with its own factors | Abundant: the factors below 90 add up to 144, which is more than 90 |
| Numbers below it sharing no factor | 24 of the numbers from 1 to 90 share no factor with 90 |
| 90 squared | 8100 |
| 90 cubed | 729000 |
| A perfect square? | No |
| A perfect cube? | No |
| A triangular number? | No |
| Nearest primes | 89 below, 97 above |
| In binary | 1011010 |
| In hexadecimal | 5A |
| As a Roman numeral | XC |
Find the factors of 90 yourself
The answers are hidden here on purpose. Type the factors one at a time; any number that divides 90 exactly counts, including 1 and 90 itself.
Frequently asked questions
- What are the factors of 90?
- 90 has 12 factors: 1, 2, 3, 5, 6, 9, 10, 15, 18, 30, 45, 90. Each one divides 90 exactly, with no remainder.
- Is 90 a prime number?
- No, 90 is a composite number. It divides exactly by 2 as well as by 1 and itself, which gives it 12 factors in total.
- What is the prime factorization of 90?
- 90 = 2 × 3 × 3 × 5, which is 2 × 3² × 5 in exponent form. Whichever pair you split 90 into first, the factor tree always ends at these same primes.
- What are the factor pairs of 90?
- There are 6: 1 × 90, 2 × 45, 3 × 30, 5 × 18, 6 × 15 and 9 × 10. Each pair is a rectangle of 90 squares, with those two numbers as its sides.
- What is the square root of 90?
- √90 is irrational. Its exact simplified form is 3√10; as a decimal it is about 9.486833, which is a rounding rather than the exact value. 90 lies between the perfect squares 81 and 100.
- How many factors does 90 have?
- 12. You can read this off the prime factorization 2 × 3² × 5 without listing anything: add one to each exponent and multiply the results.
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