1.0 KiB
1.0 KiB
| id | title | challengeType | forumTopicId | dashedName |
|---|---|---|---|---|
| 5900f51d1000cf542c51002f | Problem 433: Steps in Euclid's algorithm | 5 | 302104 | 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
```