What are the divisors of 3977?

1, 41, 97, 3977

There is a total of 4 positive divisors.
The sum of these divisors is 4116.
The arithmetic mean is 1029.

4 odd divisors

1, 41, 97, 3977

How to compute the divisors of 3977?

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 3977 by each of the numbers from 1 to 3977 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)

3977 / 1 = 3977 (the remainder is 0, so 1 is a divisor of 3977)
3977 / 2 = 1988.5 (the remainder is 1, so 2 is not a divisor of 3977)
3977 / 3 = 1325.6666666667 (the remainder is 2, so 3 is not a divisor of 3977)
...
3977 / 3976 = 1.0002515090543 (the remainder is 1, so 3976 is not a divisor of 3977)
3977 / 3977 = 1 (the remainder is 0, so 3977 is a divisor of 3977)

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 3977 (i.e. 63.06346010171). 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 $\frac{N}{d} = D$ )

3977 / 1 = 3977 (the remainder is 0, so 1 and 3977 are divisors of 3977)
3977 / 2 = 1988.5 (the remainder is 1, so 2 is not a divisor of 3977)
3977 / 3 = 1325.6666666667 (the remainder is 2, so 3 is not a divisor of 3977)
...
3977 / 62 = 64.145161290323 (the remainder is 9, so 62 is not a divisor of 3977)
3977 / 63 = 63.126984126984 (the remainder is 8, so 63 is not a divisor of 3977)