FactorsAbout 36
36 is a perfect square (6 × 6) with nine factors. Square numbers are the only numbers with an odd number of factors, because one of their factor pairs is a number with itself.
Factor pairs of 36
A factor pair is two numbers that multiply to give 36: 1 × 36, 2 × 18, 3 × 12, 4 × 9 and 6 × 6. Each pair is a way of arranging 36 things in equal rows, so 36 tiles can be laid out as a rectangle in 5 different shapes, one of them a square of side 6.
How to find the factors of 36 by hand
Divide 36 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 6, because 6 × 6 is exactly 36: any larger factor has already turned up as the partner of a smaller one.
| Try | Divides 36? | Result |
|---|---|---|
| 1 | yes | 1 × 36 |
| 2 | yes | 2 × 18 |
| 3 | yes | 3 × 12 |
| 4 | yes | 4 × 9 |
| 5 | no | remainder 1 |
| 6 | yes | 6 × 6 |
The only number that fails is 5. All the others give a pair.
Prime factorisation of 36
Divide by the smallest prime that goes in exactly, and repeat with the answer until you reach 1.
Reading down the left side: 36 = 2 × 2 × 3 × 3, which is written 22 × 32 with powers. Its prime factors are 2 and 3.
Counting the factors without listing them
Add 1 to each power in the prime factorisation and multiply the results: (2 + 1) × (2 + 1) = 9. This works because a factor of 36 can use each prime anywhere from zero times up to its full power.
The 9 factors add up to 91. Leaving out 36 itself, the rest add up to 55, which is more than 36, so 36 is an abundant number. Of its factors, 3 are odd: 1, 3 and 9.
Divisibility rules applied to 36
The quick tests agree with the factor list above.
- By 2: yes. The last digit is 6, which is even.
- By 3: yes. The digits add up to 3 + 6 = 9, which is a multiple of 3.
- By 4: yes. The last two digits make 36, and 36 ÷ 4 = 9.
- By 5: no. The last digit is 6, not 0 or 5.
- By 6: yes. A number divides by 6 when it divides by both 2 and 3. Here both tests pass.
- By 8: no. 36 ÷ 8 leaves remainder 4.
- By 9: yes. The digits add up to 9, which is a multiple of 9.
- By 10: no. The last digit is not 0.
- By 11: no. Adding alternate digits from the right gives 6 and 3. Their difference is 3, which is not 0 or a multiple of 11.
HCF and LCM with 36
36 and 32. 36 = 22 × 32 and 32 = 25. The primes they share, each with its smaller power, give the HCF: 4. Every prime with its larger power gives the LCM: 288. By division: 36 = 32 × 1 + 4; 32 = 4 × 8 + 0, so the last divisor, 4, is the HCF. Check: 4 × 288 = 1152 = 36 × 32.
36 and 40. 36 = 22 × 32 and 40 = 23 × 5. The primes they share, each with its smaller power, give the HCF: 4. Every prime with its larger power gives the LCM: 360. By division: 40 = 36 × 1 + 4; 36 = 4 × 9 + 0, so the last divisor, 4, is the HCF. Check: 4 × 360 = 1440 = 36 × 40.
Multiples of 36
Factors divide a number. Multiples are what you get by multiplying it. The first ten multiples of 36 are 36, 72, 108, 144, 180, 216, 252, 288, 324 and 360. A number has a fixed count of factors but endless multiples.
Common questions
What are the factors of 36?
The factors of 36 are 1, 2, 3, 4, 6, 9, 12, 18 and 36. There are 9 of them.
What is the prime factorisation of 36?
36 = 2 × 2 × 3 × 3.
Is 36 a perfect square?
Yes. 36 = 6 × 6, so its square root is 6.
What is the greatest factor of 36 other than itself?
18. Divide 36 by its smallest prime factor, 2, to get it.