What are the divisors of 526?

1, 2, 263, 526

There is a total of 4 positive divisors.
The sum of these divisors is 792.
The arithmetic mean is 198.

2 even divisors

2, 526

2 odd divisors

1, 263

How to compute the divisors of 526?

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 526 by each of the numbers from 1 to 526 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)

526 / 1 = 526 (the remainder is 0, so 1 is a divisor of 526)
526 / 2 = 263 (the remainder is 0, so 2 is a divisor of 526)
526 / 3 = 175.33333333333 (the remainder is 1, so 3 is not a divisor of 526)
...
526 / 525 = 1.0019047619048 (the remainder is 1, so 525 is not a divisor of 526)
526 / 526 = 1 (the remainder is 0, so 526 is a divisor of 526)

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 526 (i.e. 22.934689882359). 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 $\frac{N}{d} = D$ )

526 / 1 = 526 (the remainder is 0, so 1 and 526 are divisors of 526)
526 / 2 = 263 (the remainder is 0, so 2 and 263 are divisors of 526)
526 / 3 = 175.33333333333 (the remainder is 1, so 3 is not a divisor of 526)
...
526 / 21 = 25.047619047619 (the remainder is 1, so 21 is not a divisor of 526)
526 / 22 = 23.909090909091 (the remainder is 20, so 22 is not a divisor of 526)