# Project Euler #21: Amicable numbers

This article is part of a series where I'll be diving head first into the Project Euler puzzles. I want to document the challenge of solving such a puzzle and how I got to the answer. I want to prefix this by stating that I can't cheat for any of these challenges; with that I mean I can't look up any other implementations online. After the implementation, I will validate the answer by using this document or a similar sheet.

### Introduction

Amicable numbers are numbers where d(a) = b and d(b) = a and a != b. To determine d(N), you find all divisors for N and sum them. You take that sum and check if d(Sum) returns the same N, if it does, it’s a pair. If it doesn’t, it isn’t a pair. Find and sum up all the pairs below 10.000.

To determine the sum of divisors to a number you can do the following:

``````fn sum_divisors(i: u16) -> u16 {
let sqrt = (i as f64).sqrt() as u16;
let mut total_div = 1; // Every number is divisible by 1
for n in 2..sqrt {
if i % n == 0 {
total_div += n + (i / n);
}
}
total_div
}

#[test]
fn test_divisors() {
assert_eq!(divisors(220), 284);
assert_eq!(divisors(60), 108);
}
``````

With that in place, an amicable pair can be thought of like this:

``````sum_divisors(sum_divisors(n)) == n;
``````

Upon implementing this looks like:

``````fn problem_21() -> u16 {
(0..10_000)
.filter(|&n| sum_divisors(sum_divisors(n)) == n)
.sum()
}
``````

The code returns 40279 and upon checking the answer (31626), it seems like I did something wrong. I’m 8653 off, which is quite a lot. I believe I’m missing the part where a != b. Let’s be actually correct this time about what amicable means and abstract it to a function returning a boolean:

``````fn amicable(i: u16) -> bool {
let n = sum_divisors(i);

n != i && sum_divisors(n) == i
}

#[test]
fn test_amicable() {
assert_eq!(amicable(1), false);
assert_eq!(amicable(220), true);
}
``````

That seems to be the fix for my problem and it returns the correct result of 31626. The full code being:

``````fn sum_divisors(i: u16) -> u16 {
let sqrt = (i as f64).sqrt() as u16;
let mut total_div = 1;
for n in 2..sqrt {
if i % n == 0 {
total_div += n + (i / n);
}
}
total_div
}

#[test]
fn test_divisors() {
assert_eq!(sum_divisors(220), 284);
assert_eq!(sum_divisors(60), 108);
assert_eq!(sum_divisors(sum_divisors(220)), 220);
}

fn amicable(i: u16) -> bool {
let n = sum_divisors(i);

n != i && sum_divisors(n) == i
}

#[test]
fn test_amicable() {
assert_eq!(amicable(1), false);
assert_eq!(amicable(220), true);
}

fn problem_21() -> u16 {
(0..10_000)
.filter(|&n| amicable(n))
.sum()
}

#[test]
fn test_problem_21() {
assert_eq!(problem_21(), 31626);
}
``````

The full solution is available on GitHub.