What are the divisors of 162?

1, 2, 3, 6, 9, 18, 27, 54, 81, 162

There is a total of 10 positive divisors.
The sum of these divisors is 363.
The arithmetic mean is 36.3.

5 even divisors

2, 6, 18, 54, 162

5 odd divisors

1, 3, 9, 27, 81

How to compute the divisors of 162?

A number N is said to be divisible by a number M (with M non-zero) if, when we divide N by M, the remainder of the division is zero.

$N mod M = 0$

Brute force algorithm

We could start by using a brute-force method which would involve dividing 162 by each of the numbers from 1 to 162 to determine which ones have a remainder equal to 0.

$Remainder = N - (M \times ⌊ \frac{N}{M} ⌋)$

(where $⌊ \frac{N}{M} ⌋$ is the integer part of the quotient)

162 / 1 = 162 (the remainder is 0, so 1 is a divisor of 162)
162 / 2 = 81 (the remainder is 0, so 2 is a divisor of 162)
162 / 3 = 54 (the remainder is 0, so 3 is a divisor of 162)
...
162 / 161 = 1.0062111801242 (the remainder is 1, so 161 is not a divisor of 162)
162 / 162 = 1 (the remainder is 0, so 162 is a divisor of 162)

Improved algorithm using square-root

However, there is another slightly better approach that reduces the number of iterations by testing only integers less than or equal to the square root of 162 (i.e. 12.727922061358). Indeed, if a number N has a divisor D greater than its square root, then there is necessarily a smaller divisor d such that:

$D \times d = N$

(thus, if $\frac{N}{D} = d$ , then $\frac{N}{d} = D$ )

162 / 1 = 162 (the remainder is 0, so 1 and 162 are divisors of 162)
162 / 2 = 81 (the remainder is 0, so 2 and 81 are divisors of 162)
162 / 3 = 54 (the remainder is 0, so 3 and 54 are divisors of 162)
...
162 / 11 = 14.727272727273 (the remainder is 8, so 11 is not a divisor of 162)
162 / 12 = 13.5 (the remainder is 6, so 12 is not a divisor of 162)