Factors of 72
72 has 12 factors: 1, 2, 3, 4, 6, 8, 9, 12, 18, 24, 36 and 72. This page shows the pairs that multiply to make 72, its factor tree, the divisibility rules it passes, its multiples and its square root, with the working for each one.
All 12 factors of 72
12346891218243672
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 × 72, 2 × 36, 3 × 24, 4 × 18, 6 × 12 and 8 × 9
- Prime factorization
- 72 = 2³ × 3²
Why these are the factors of 72
Two numbers form a factor pair of 72 when 72 squares can be arranged into a rectangle with those side lengths and none left over. 72 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 72
Split 72 into any two factors, then keep splitting until only primes are left. The bottom row gives 72 = 2 × 2 × 2 × 3 × 3. 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.
72 = 2 × 2 × 2 × 3 × 3
However you split it, the bottom row stays the same. Every number has exactly one prime factorization.
Divisibility rules, checked against 72
Most factors of 72 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 2. | Yes: 72 ÷ 2 = 36 |
| 3 | The digits add up to a multiple of 3. Here they add to 9. | Yes: 72 ÷ 3 = 24 |
| 4 | The last two digits form a multiple of 4. Here they are 72. | Yes: 72 ÷ 4 = 18 |
| 5 | The last digit is 0 or 5. Here it is 2. | No: 72 ÷ 5 leaves remainder 2 |
| 6 | It passes both the 2 rule and the 3 rule. | Yes: 72 ÷ 6 = 12 |
| 7 | Double the last digit and subtract it from the rest. Here that gives 3. | No: 72 ÷ 7 leaves remainder 2 |
| 8 | The last three digits form a multiple of 8. Here they are 72. | Yes: 72 ÷ 8 = 9 |
| 9 | The digits add up to a multiple of 9. Here they add to 9. | Yes: 72 ÷ 9 = 8 |
| 10 | The last digit is 0. Here it is 2. | No: 72 ÷ 10 leaves remainder 2 |
| 11 | Add and subtract the digits alternately from the right. Here that gives -5. | No: 72 ÷ 11 leaves remainder 6 |
Multiples of 72
Multiples run in the other direction: they are the numbers 72 divides into exactly. The list starts 72, 144, 216, 288, 360, 432 and goes on forever, growing by 72 each step. Every multiple of 72 has 72 in its own factor list.
The square root of 72
√72 is not a whole number, because 72 sits between the perfect squares 64 and 81. Its exact simplified form is 6√2: the square factor moves out from under the root sign. The decimal 8.485281 is a rounded approximation, not the exact value.
Everything else about 72
| Even or odd | Even |
|---|---|
| Prime or composite | Composite |
| Number of factors | 12, which matches the rule from 2³ × 3²: add one to each exponent and multiply. |
| Factors added together | 195 |
| Compared with its own factors | Abundant: the factors below 72 add up to 123, which is more than 72 |
| Numbers below it sharing no factor | 24 of the numbers from 1 to 72 share no factor with 72 |
| 72 squared | 5184 |
| 72 cubed | 373248 |
| A perfect square? | No |
| A perfect cube? | No |
| A triangular number? | No |
| Nearest primes | 71 below, 73 above |
| In binary | 1001000 |
| In hexadecimal | 48 |
| As a Roman numeral | LXXII |
Find the factors of 72 yourself
The answers are hidden here on purpose. Type the factors one at a time; any number that divides 72 exactly counts, including 1 and 72 itself.
Frequently asked questions
- What are the factors of 72?
- 72 has 12 factors: 1, 2, 3, 4, 6, 8, 9, 12, 18, 24, 36, 72. Each one divides 72 exactly, with no remainder.
- Is 72 a prime number?
- No, 72 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 72?
- 72 = 2 × 2 × 2 × 3 × 3, which is 2³ × 3² in exponent form. Whichever pair you split 72 into first, the factor tree always ends at these same primes.
- What are the factor pairs of 72?
- There are 6: 1 × 72, 2 × 36, 3 × 24, 4 × 18, 6 × 12 and 8 × 9. Each pair is a rectangle of 72 squares, with those two numbers as its sides.
- What is the square root of 72?
- √72 is irrational. Its exact simplified form is 6√2; as a decimal it is about 8.485281, which is a rounding rather than the exact value. 72 lies between the perfect squares 64 and 81.
- How many factors does 72 have?
- 12. You can read this off the prime factorization 2³ × 3² without listing anything: add one to each exponent and multiply the results.
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