# Is 864 a prime number?

It is possible to find out using mathematical methods whether a given integer is a prime number or not.

For 864, the answer is: No, 864 is not a prime number.

The list of all positive divisors (i.e., the list of all integers that divide 864) is as follows: 1, 2, 3, 4, 6, 8, 9, 12, 16, 18, 24, 27, 32, 36, 48, 54, 72, 96, 108, 144, 216, 288, 432, 864.

For 864 to be a prime number, it would have been required that 864 has only two divisors, i.e., itself and 1.

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As a consequence:

- 864 is a multiple of 1
- 864 is a multiple of 2
- 864 is a multiple of 3
- 864 is a multiple of 4
- 864 is a multiple of 6
- 864 is a multiple of 8
- 864 is a multiple of 9
- 864 is a multiple of 12
- 864 is a multiple of 16
- 864 is a multiple of 18
- 864 is a multiple of 24
- 864 is a multiple of 27
- 864 is a multiple of 32
- 864 is a multiple of 36
- 864 is a multiple of 48
- 864 is a multiple of 54
- 864 is a multiple of 72
- 864 is a multiple of 96
- 864 is a multiple of 108
- 864 is a multiple of 144
- 864 is a multiple of 216
- 864 is a multiple of 288
- 864 is a multiple of 432

For 864 to be a prime number, it would have been required that 864 has only two divisors, i.e., itself and 1.

### Is 864 a deficient number?

No, 864 is not a deficient number: to be deficient, 864 should have been such that 864 is larger than the sum of its proper divisors, i.e., the divisors of 864 without 864 itself (that is 1 + 2 + 3 + 4 + 6 + 8 + 9 + 12 + 16 + 18 + 24 + 27 + 32 + 36 + 48 + 54 + 72 + 96 + 108 + 144 + 216 + 288 + 432 = 1 656).

In fact, 864 is an abundant number; 864 is strictly smaller than the sum of its proper divisors (that is 1 + 2 + 3 + 4 + 6 + 8 + 9 + 12 + 16 + 18 + 24 + 27 + 32 + 36 + 48 + 54 + 72 + 96 + 108 + 144 + 216 + 288 + 432 = 1 656). The smallest abundant number is 12.