What are the divisors of 3721?

1, 61, 3721

There is a total of 3 positive divisors.
The sum of these divisors is 3783.
The arithmetic mean is 1261.

3 odd divisors

1, 61, 3721

How to compute the divisors of 3721?

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 3721 by each of the numbers from 1 to 3721 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)

3721 / 1 = 3721 (the remainder is 0, so 1 is a divisor of 3721)
3721 / 2 = 1860.5 (the remainder is 1, so 2 is not a divisor of 3721)
3721 / 3 = 1240.3333333333 (the remainder is 1, so 3 is not a divisor of 3721)
...
3721 / 3720 = 1.0002688172043 (the remainder is 1, so 3720 is not a divisor of 3721)
3721 / 3721 = 1 (the remainder is 0, so 3721 is a divisor of 3721)

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 3721 (i.e. 61). 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 $\frac{N}{d} = D$ )

3721 / 1 = 3721 (the remainder is 0, so 1 and 3721 are divisors of 3721)
3721 / 2 = 1860.5 (the remainder is 1, so 2 is not a divisor of 3721)
3721 / 3 = 1240.3333333333 (the remainder is 1, so 3 is not a divisor of 3721)
...
3721 / 60 = 62.016666666667 (the remainder is 1, so 60 is not a divisor of 3721)
3721 / 61 = 61 (the remainder is 0, so 61 and 61 are divisors of 3721)