What are the divisors of 35?

1, 5, 7, 35

There is a total of 4 positive divisors.
The sum of these divisors is 48.
The arithmetic mean is 12.

4 odd divisors

1, 5, 7, 35

How to compute the divisors of 35?

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 35 by each of the numbers from 1 to 35 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)

35 / 1 = 35 (the remainder is 0, so 1 is a divisor of 35)
35 / 2 = 17.5 (the remainder is 1, so 2 is not a divisor of 35)
35 / 3 = 11.666666666667 (the remainder is 2, so 3 is not a divisor of 35)
...
35 / 34 = 1.0294117647059 (the remainder is 1, so 34 is not a divisor of 35)
35 / 35 = 1 (the remainder is 0, so 35 is a divisor of 35)

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 35 (i.e. 5.9160797830996). 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 $\frac{N}{d} = D$ )

35 / 1 = 35 (the remainder is 0, so 1 and 35 are divisors of 35)
35 / 2 = 17.5 (the remainder is 1, so 2 is not a divisor of 35)
35 / 3 = 11.666666666667 (the remainder is 2, so 3 is not a divisor of 35)
...
35 / 4 = 8.75 (the remainder is 3, so 4 is not a divisor of 35)
35 / 5 = 7 (the remainder is 0, so 5 and 7 are divisors of 35)