What are the divisors of 3497?

1, 13, 269, 3497

There is a total of 4 positive divisors.
The sum of these divisors is 3780.
The arithmetic mean is 945.

4 odd divisors

1, 13, 269, 3497

How to compute the divisors of 3497?

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 3497 by each of the numbers from 1 to 3497 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)

3497 / 1 = 3497 (the remainder is 0, so 1 is a divisor of 3497)
3497 / 2 = 1748.5 (the remainder is 1, so 2 is not a divisor of 3497)
3497 / 3 = 1165.6666666667 (the remainder is 2, so 3 is not a divisor of 3497)
...
3497 / 3496 = 1.0002860411899 (the remainder is 1, so 3496 is not a divisor of 3497)
3497 / 3497 = 1 (the remainder is 0, so 3497 is a divisor of 3497)

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 3497 (i.e. 59.13543776789). 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 $\frac{N}{d} = D$ )

3497 / 1 = 3497 (the remainder is 0, so 1 and 3497 are divisors of 3497)
3497 / 2 = 1748.5 (the remainder is 1, so 2 is not a divisor of 3497)
3497 / 3 = 1165.6666666667 (the remainder is 2, so 3 is not a divisor of 3497)
...
3497 / 58 = 60.293103448276 (the remainder is 17, so 58 is not a divisor of 3497)
3497 / 59 = 59.271186440678 (the remainder is 16, so 59 is not a divisor of 3497)