73 lines
1.1 KiB
Markdown
73 lines
1.1 KiB
Markdown
|
---
|
||
|
id: 5900f5311000cf542c510042
|
||
|
challengeType: 5
|
||
|
title: 'Problem 451: Modular inverses'
|
||
|
forumTopicId: 302124
|
||
|
---
|
||
|
|
||
|
## Description
|
||
|
<section id='description'>
|
||
|
Consider the number 15.
|
||
|
There are eight positive numbers less than 15 which are coprime to 15: 1, 2, 4, 7, 8, 11, 13, 14.
|
||
|
The modular inverses of these numbers modulo 15 are: 1, 8, 4, 13, 2, 11, 7, 14
|
||
|
because
|
||
|
1*1 mod 15=1
|
||
|
2*8=16 mod 15=1
|
||
|
4*4=16 mod 15=1
|
||
|
7*13=91 mod 15=1
|
||
|
11*11=121 mod 15=1
|
||
|
14*14=196 mod 15=1
|
||
|
|
||
|
Let I(n) be the largest positive number m smaller than n-1 such that the modular inverse of m modulo n equals m itself.
|
||
|
So I(15)=11.
|
||
|
Also I(100)=51 and I(7)=1.
|
||
|
|
||
|
Find ∑I(n) for 3≤n≤2·107
|
||
|
</section>
|
||
|
|
||
|
## Instructions
|
||
|
<section id='instructions'>
|
||
|
|
||
|
</section>
|
||
|
|
||
|
## Tests
|
||
|
<section id='tests'>
|
||
|
|
||
|
```yml
|
||
|
tests:
|
||
|
- text: <code>euler451()</code> should return 153651073760956.
|
||
|
testString: assert.strictEqual(euler451(), 153651073760956);
|
||
|
|
||
|
```
|
||
|
|
||
|
</section>
|
||
|
|
||
|
## Challenge Seed
|
||
|
<section id='challengeSeed'>
|
||
|
|
||
|
<div id='js-seed'>
|
||
|
|
||
|
```js
|
||
|
function euler451() {
|
||
|
|
||
|
return true;
|
||
|
}
|
||
|
|
||
|
euler451();
|
||
|
```
|
||
|
|
||
|
</div>
|
||
|
|
||
|
|
||
|
|
||
|
</section>
|
||
|
|
||
|
## Solution
|
||
|
<section id='solution'>
|
||
|
|
||
|
```js
|
||
|
// solution required
|
||
|
```
|
||
|
|
||
|
</section>
|