What are the divisors of 1421?

1, 7, 29, 49, 203, 1421

There is a total of 6 positive divisors.
The sum of these divisors is 1710.
The arithmetic mean is 285.

6 odd divisors

1, 7, 29, 49, 203, 1421

How to compute the divisors of 1421?

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 1421 by each of the numbers from 1 to 1421 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)

1421 / 1 = 1421 (the remainder is 0, so 1 is a divisor of 1421)
1421 / 2 = 710.5 (the remainder is 1, so 2 is not a divisor of 1421)
1421 / 3 = 473.66666666667 (the remainder is 2, so 3 is not a divisor of 1421)
...
1421 / 1420 = 1.0007042253521 (the remainder is 1, so 1420 is not a divisor of 1421)
1421 / 1421 = 1 (the remainder is 0, so 1421 is a divisor of 1421)

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 1421 (i.e. 37.696153649942). 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 $\frac{N}{d} = D$ )

1421 / 1 = 1421 (the remainder is 0, so 1 and 1421 are divisors of 1421)
1421 / 2 = 710.5 (the remainder is 1, so 2 is not a divisor of 1421)
1421 / 3 = 473.66666666667 (the remainder is 2, so 3 is not a divisor of 1421)
...
1421 / 36 = 39.472222222222 (the remainder is 17, so 36 is not a divisor of 1421)
1421 / 37 = 38.405405405405 (the remainder is 15, so 37 is not a divisor of 1421)