61 lines
958 B
Markdown
61 lines
958 B
Markdown
---
|
|
id: 5900f5311000cf542c510042
|
|
title: 'Problem 451: Modular inverses'
|
|
challengeType: 5
|
|
forumTopicId: 302124
|
|
dashedName: problem-451-modular-inverses
|
|
---
|
|
|
|
# --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
|
|
|
|
# --hints--
|
|
|
|
`euler451()` should return 153651073760956.
|
|
|
|
```js
|
|
assert.strictEqual(euler451(), 153651073760956);
|
|
```
|
|
|
|
# --seed--
|
|
|
|
## --seed-contents--
|
|
|
|
```js
|
|
function euler451() {
|
|
|
|
return true;
|
|
}
|
|
|
|
euler451();
|
|
```
|
|
|
|
# --solutions--
|
|
|
|
```js
|
|
// solution required
|
|
```
|