54 lines
1.0 KiB
Markdown
54 lines
1.0 KiB
Markdown
---
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id: 5900f51d1000cf542c51002f
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title: 'Problem 433: Steps in Euclid''s algorithm'
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challengeType: 5
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forumTopicId: 302104
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dashedName: problem-433-steps-in-euclids-algorithm
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---
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# --description--
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Let $E(x_0, y_0)$ be the number of steps it takes to determine the greatest common divisor of $x_0$ and $y_0$ with Euclid's algorithm. More formally:
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$$\begin{align}
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& x_1 = y_0, y_1 = x_0\bmod y_0 \\\\
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& x_n = y_{n - 1}, y_n = x_{n - 1}\bmod y_{n - 1}
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\end{align}$$
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$E(x_0, y_0)$ is the smallest $n$ such that $y_n = 0$.
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We have $E(1, 1) = 1$, $E(10, 6) = 3$ and $E(6, 10) = 4$.
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Define $S(N)$ as the sum of $E(x, y)$ for $1 ≤ x$, $y ≤ N$.
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We have $S(1) = 1$, $S(10) = 221$ and $S(100) = 39\\,826$.
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Find $S(5 \times {10}^6)$.
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# --hints--
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`stepsInEuclidsAlgorithm()` should return `326624372659664`.
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```js
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assert.strictEqual(stepsInEuclidsAlgorithm(), 326624372659664);
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```
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# --seed--
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## --seed-contents--
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```js
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function stepsInEuclidsAlgorithm() {
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return true;
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}
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stepsInEuclidsAlgorithm();
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```
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# --solutions--
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```js
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// solution required
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```
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