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