What are the divisors of 1954?

1, 2, 977, 1954

There is a total of 4 positive divisors.
The sum of these divisors is 2934.
The arithmetic mean is 733.5.

2 even divisors

2, 1954

2 odd divisors

1, 977

How to compute the divisors of 1954?

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 1954 by each of the numbers from 1 to 1954 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)

1954 / 1 = 1954 (the remainder is 0, so 1 is a divisor of 1954)
1954 / 2 = 977 (the remainder is 0, so 2 is a divisor of 1954)
1954 / 3 = 651.33333333333 (the remainder is 1, so 3 is not a divisor of 1954)
...
1954 / 1953 = 1.0005120327701 (the remainder is 1, so 1953 is not a divisor of 1954)
1954 / 1954 = 1 (the remainder is 0, so 1954 is a divisor of 1954)

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 1954 (i.e. 44.204072210601). 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 $\frac{N}{d} = D$ )

1954 / 1 = 1954 (the remainder is 0, so 1 and 1954 are divisors of 1954)
1954 / 2 = 977 (the remainder is 0, so 2 and 977 are divisors of 1954)
1954 / 3 = 651.33333333333 (the remainder is 1, so 3 is not a divisor of 1954)
...
1954 / 43 = 45.441860465116 (the remainder is 19, so 43 is not a divisor of 1954)
1954 / 44 = 44.409090909091 (the remainder is 18, so 44 is not a divisor of 1954)