What are the divisors of 1601?

1, 1601

There is a total of 2 positive divisors.
The sum of these divisors is 1602.
The arithmetic mean is 801.

2 odd divisors

1, 1601

How to compute the divisors of 1601?

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 1601 by each of the numbers from 1 to 1601 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)

1601 / 1 = 1601 (the remainder is 0, so 1 is a divisor of 1601)
1601 / 2 = 800.5 (the remainder is 1, so 2 is not a divisor of 1601)
1601 / 3 = 533.66666666667 (the remainder is 2, so 3 is not a divisor of 1601)
...
1601 / 1600 = 1.000625 (the remainder is 1, so 1600 is not a divisor of 1601)
1601 / 1601 = 1 (the remainder is 0, so 1601 is a divisor of 1601)

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 1601 (i.e. 40.012498047485). 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 $\frac{N}{d} = D$ )

1601 / 1 = 1601 (the remainder is 0, so 1 and 1601 are divisors of 1601)
1601 / 2 = 800.5 (the remainder is 1, so 2 is not a divisor of 1601)
1601 / 3 = 533.66666666667 (the remainder is 2, so 3 is not a divisor of 1601)
...
1601 / 39 = 41.051282051282 (the remainder is 2, so 39 is not a divisor of 1601)
1601 / 40 = 40.025 (the remainder is 1, so 40 is not a divisor of 1601)