What are the divisors of 143?

1, 11, 13, 143

There is a total of 4 positive divisors.
The sum of these divisors is 168.
The arithmetic mean is 42.

4 odd divisors

1, 11, 13, 143

How to compute the divisors of 143?

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 143 by each of the numbers from 1 to 143 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)

143 / 1 = 143 (the remainder is 0, so 1 is a divisor of 143)
143 / 2 = 71.5 (the remainder is 1, so 2 is not a divisor of 143)
143 / 3 = 47.666666666667 (the remainder is 2, so 3 is not a divisor of 143)
...
143 / 142 = 1.0070422535211 (the remainder is 1, so 142 is not a divisor of 143)
143 / 143 = 1 (the remainder is 0, so 143 is a divisor of 143)

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 143 (i.e. 11.958260743101). 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 $\frac{N}{d} = D$ )

143 / 1 = 143 (the remainder is 0, so 1 and 143 are divisors of 143)
143 / 2 = 71.5 (the remainder is 1, so 2 is not a divisor of 143)
143 / 3 = 47.666666666667 (the remainder is 2, so 3 is not a divisor of 143)
...
143 / 10 = 14.3 (the remainder is 3, so 10 is not a divisor of 143)
143 / 11 = 13 (the remainder is 0, so 11 and 13 are divisors of 143)