FactorsAbout 64
A chessboard has 64 squares. 64 is both a perfect square (8 × 8) and a perfect cube (4 × 4 × 4), the smallest number above 1 that is both.
Factor pairs of 64
A factor pair is two numbers that multiply to give 64: 1 × 64, 2 × 32, 4 × 16 and 8 × 8. Each pair is a way of arranging 64 things in equal rows, so 64 tiles can be laid out as a rectangle in 4 different shapes, one of them a square of side 8.
How to find the factors of 64 by hand
Divide 64 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 8, because 8 × 8 is exactly 64: any larger factor has already turned up as the partner of a smaller one.
| Try | Divides 64? | Result |
|---|---|---|
| 1 | yes | 1 × 64 |
| 2 | yes | 2 × 32 |
| 3 | no | remainder 1 |
| 4 | yes | 4 × 16 |
| 5 | no | remainder 4 |
| 6 | no | remainder 4 |
| 7 | no | remainder 1 |
| 8 | yes | 8 × 8 |
The numbers that fail are 3, 5, 6 and 7. All the others give a pair.
Prime factorisation of 64
Divide by the smallest prime that goes in exactly, and repeat with the answer until you reach 1.
Reading down the left side: 64 = 2 × 2 × 2 × 2 × 2 × 2, which is written 26 with powers. It has only one prime factor, 2.
Counting the factors without listing them
Add 1 to each power in the prime factorisation and multiply the results: (6 + 1) = 7. This works because a factor of 64 can use each prime anywhere from zero times up to its full power.
The 7 factors add up to 127. Leaving out 64 itself, the rest add up to 63, which is less than 64, so 64 is a deficient number. Of its factors, only one is odd: 1.
Divisibility rules applied to 64
The quick tests agree with the factor list above.
- By 2: yes. The last digit is 4, which is even.
- By 3: no. The digits add up to 6 + 4 = 10, which is not a multiple of 3.
- By 4: yes. The last two digits make 64, and 64 ÷ 4 = 16.
- By 5: no. The last digit is 4, not 0 or 5.
- By 6: no. A number divides by 6 when it divides by both 2 and 3. Here the test for 3 fails.
- By 8: yes. 64 ÷ 8 = 8.
- By 9: no. The digits add up to 10, which is not a multiple of 9.
- By 10: no. The last digit is not 0.
- By 11: no. Adding alternate digits from the right gives 4 and 6. Their difference is 2, which is not 0 or a multiple of 11.
HCF and LCM with 64
64 and 63. 64 = 26 and 63 = 32 × 7. The primes they share, each with its smaller power, give the HCF: 1. Every prime with its larger power gives the LCM: 4032. By division: 64 = 63 × 1 + 1; 63 = 1 × 63 + 0, so the last divisor, 1, is the HCF. Check: 1 × 4032 = 4032 = 64 × 63.
64 and 72. 64 = 26 and 72 = 23 × 32. The primes they share, each with its smaller power, give the HCF: 8. Every prime with its larger power gives the LCM: 576. By division: 72 = 64 × 1 + 8; 64 = 8 × 8 + 0, so the last divisor, 8, is the HCF. Check: 8 × 576 = 4608 = 64 × 72.
Multiples of 64
Factors divide a number. Multiples are what you get by multiplying it. The first ten multiples of 64 are 64, 128, 192, 256, 320, 384, 448, 512, 576 and 640. A number has a fixed count of factors but endless multiples.
Common questions
What are the factors of 64?
The factors of 64 are 1, 2, 4, 8, 16, 32 and 64. There are 7 of them.
What is the prime factorisation of 64?
64 = 2 × 2 × 2 × 2 × 2 × 2.
Is 64 a perfect square?
Yes. 64 = 8 × 8, so its square root is 8.
What is the greatest factor of 64 other than itself?
32. Divide 64 by its smallest prime factor, 2, to get it.