What are the divisors of 1948?

1, 2, 4, 487, 974, 1948

There is a total of 6 positive divisors.
The sum of these divisors is 3416.
The arithmetic mean is 569.33333333333.

4 even divisors

2, 4, 974, 1948

2 odd divisors

1, 487

How to compute the divisors of 1948?

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 1948 by each of the numbers from 1 to 1948 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)

1948 / 1 = 1948 (the remainder is 0, so 1 is a divisor of 1948)
1948 / 2 = 974 (the remainder is 0, so 2 is a divisor of 1948)
1948 / 3 = 649.33333333333 (the remainder is 1, so 3 is not a divisor of 1948)
...
1948 / 1947 = 1.0005136106831 (the remainder is 1, so 1947 is not a divisor of 1948)
1948 / 1948 = 1 (the remainder is 0, so 1948 is a divisor of 1948)

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 1948 (i.e. 44.136152981428). 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 $\frac{N}{d} = D$ )

1948 / 1 = 1948 (the remainder is 0, so 1 and 1948 are divisors of 1948)
1948 / 2 = 974 (the remainder is 0, so 2 and 974 are divisors of 1948)
1948 / 3 = 649.33333333333 (the remainder is 1, so 3 is not a divisor of 1948)
...
1948 / 43 = 45.302325581395 (the remainder is 13, so 43 is not a divisor of 1948)
1948 / 44 = 44.272727272727 (the remainder is 12, so 44 is not a divisor of 1948)