2018-09-30 22:01:58 +00:00
---
id: 5900f51d1000cf542c51002f
2018-10-20 18:02:47 +00:00
title: 'Problem 433: Steps in Euclid''s algorithm'
2020-11-27 18:02:05 +00:00
challengeType: 5
2019-08-05 16:17:33 +00:00
forumTopicId: 302104
2021-01-13 02:31:00 +00:00
dashedName: problem-433-steps-in-euclids-algorithm
2018-09-30 22:01:58 +00:00
---
2020-11-27 18:02:05 +00:00
# --description--
2018-09-30 22:01:58 +00:00
Let E(x0, y0) be the number of steps it takes to determine the greatest common divisor of x0 and y0 with Euclid's algorithm. More formally:x1 = y0, y1 = x0 mod y0xn = yn-1, yn = xn-1 mod yn-1
2020-11-27 18:02:05 +00:00
E(x0, y0) is the smallest n such that yn = 0.
2018-09-30 22:01:58 +00:00
We have E(1,1) = 1, E(10,6) = 3 and E(6,10) = 4.
2020-11-27 18:02:05 +00:00
Define S(N) as the sum of E(x,y) for 1 ≤ x,y ≤ N. We have S(1) = 1, S(10) = 221 and S(100) = 39826.
2018-09-30 22:01:58 +00:00
Find S(5·106).
2020-11-27 18:02:05 +00:00
# --hints--
2018-09-30 22:01:58 +00:00
2020-11-27 18:02:05 +00:00
`euler433()` should return 326624372659664.
2018-09-30 22:01:58 +00:00
2020-11-27 18:02:05 +00:00
```js
assert.strictEqual(euler433(), 326624372659664);
2018-09-30 22:01:58 +00:00
```
2020-11-27 18:02:05 +00:00
# --seed--
2018-09-30 22:01:58 +00:00
2020-11-27 18:02:05 +00:00
## --seed-contents--
2018-09-30 22:01:58 +00:00
```js
function euler433() {
2020-09-15 16:57:40 +00:00
2018-09-30 22:01:58 +00:00
return true;
}
euler433();
```
2020-11-27 18:02:05 +00:00
# --solutions--
2018-09-30 22:01:58 +00:00
```js
// solution required
```