What are the divisors of 3136?

1, 2, 4, 7, 8, 14, 16, 28, 32, 49, 56, 64, 98, 112, 196, 224, 392, 448, 784, 1568, 3136

There is a total of 21 positive divisors.
The sum of these divisors is 7239.
The arithmetic mean is 344.71428571429.

18 even divisors

2, 4, 8, 14, 16, 28, 32, 56, 64, 98, 112, 196, 224, 392, 448, 784, 1568, 3136

3 odd divisors

1, 7, 49

How to compute the divisors of 3136?

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 3136 by each of the numbers from 1 to 3136 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)

3136 / 1 = 3136 (the remainder is 0, so 1 is a divisor of 3136)
3136 / 2 = 1568 (the remainder is 0, so 2 is a divisor of 3136)
3136 / 3 = 1045.3333333333 (the remainder is 1, so 3 is not a divisor of 3136)
...
3136 / 3135 = 1.0003189792663 (the remainder is 1, so 3135 is not a divisor of 3136)
3136 / 3136 = 1 (the remainder is 0, so 3136 is a divisor of 3136)

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 3136 (i.e. 56). 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 $\frac{N}{d} = D$ )

3136 / 1 = 3136 (the remainder is 0, so 1 and 3136 are divisors of 3136)
3136 / 2 = 1568 (the remainder is 0, so 2 and 1568 are divisors of 3136)
3136 / 3 = 1045.3333333333 (the remainder is 1, so 3 is not a divisor of 3136)
...
3136 / 55 = 57.018181818182 (the remainder is 1, so 55 is not a divisor of 3136)
3136 / 56 = 56 (the remainder is 0, so 56 and 56 are divisors of 3136)