FactorsAbout 81
81 is 9 × 9 and also 3 × 3 × 3 × 3. As the fourth power of a prime it has exactly five factors.
Factor pairs of 81
A factor pair is two numbers that multiply to give 81: 1 × 81, 3 × 27 and 9 × 9. Each pair is a way of arranging 81 things in equal rows, so 81 tiles can be laid out as a rectangle in 3 different shapes, one of them a square of side 9.
How to find the factors of 81 by hand
Divide 81 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 9 × 9 is exactly 81: any larger factor has already turned up as the partner of a smaller one.
| Try | Divides 81? | Result |
|---|---|---|
| 1 | yes | 1 × 81 |
| 2 | no | remainder 1 |
| 3 | yes | 3 × 27 |
| 4 | no | remainder 1 |
| 5 | no | remainder 1 |
| 6 | no | remainder 3 |
| 7 | no | remainder 4 |
| 8 | no | remainder 1 |
| 9 | yes | 9 × 9 |
The numbers that fail are 2, 4, 5, 6, 7 and 8. All the others give a pair.
Prime factorisation of 81
Divide by the smallest prime that goes in exactly, and repeat with the answer until you reach 1.
Reading down the left side: 81 = 3 × 3 × 3 × 3, which is written 34 with powers. It has only one prime factor, 3.
Counting the factors without listing them
Add 1 to each power in the prime factorisation and multiply the results: (4 + 1) = 5. This works because a factor of 81 can use each prime anywhere from zero times up to its full power.
The 5 factors add up to 121. Leaving out 81 itself, the rest add up to 40, which is less than 81, so 81 is a deficient number. Of its factors, 5 are odd: 1, 3, 9, 27 and 81.
Divisibility rules applied to 81
The quick tests agree with the factor list above.
- By 2: no. The last digit is 1, which is odd.
- By 3: yes. The digits add up to 8 + 1 = 9, which is a multiple of 3.
- By 4: no. The last two digits make 81, and 81 ÷ 4 leaves remainder 1.
- By 5: no. The last digit is 1, not 0 or 5.
- By 6: no. A number divides by 6 when it divides by both 2 and 3. Here the test for 2 fails.
- By 8: no. 81 ÷ 8 leaves remainder 1.
- 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 1 and 8. Their difference is 7, which is not 0 or a multiple of 11.
HCF and LCM with 81
81 and 80. 81 = 34 and 80 = 24 × 5. The primes they share, each with its smaller power, give the HCF: 1. Every prime with its larger power gives the LCM: 6480. By division: 81 = 80 × 1 + 1; 80 = 1 × 80 + 0, so the last divisor, 1, is the HCF. Check: 1 × 6480 = 6480 = 81 × 80.
81 and 84. 81 = 34 and 84 = 22 × 3 × 7. The primes they share, each with its smaller power, give the HCF: 3. Every prime with its larger power gives the LCM: 2268. By division: 84 = 81 × 1 + 3; 81 = 3 × 27 + 0, so the last divisor, 3, is the HCF. Check: 3 × 2268 = 6804 = 81 × 84.
Multiples of 81
Factors divide a number. Multiples are what you get by multiplying it. The first ten multiples of 81 are 81, 162, 243, 324, 405, 486, 567, 648, 729 and 810. A number has a fixed count of factors but endless multiples.
Common questions
What are the factors of 81?
The factors of 81 are 1, 3, 9, 27 and 81. There are 5 of them.
What is the prime factorisation of 81?
81 = 3 × 3 × 3 × 3.
Is 81 a perfect square?
Yes. 81 = 9 × 9, so its square root is 9.
What is the greatest factor of 81 other than itself?
27. Divide 81 by its smallest prime factor, 3, to get it.