51 lines
985 B
Markdown
51 lines
985 B
Markdown
---
|
|
id: 5900f52c1000cf542c51003e
|
|
title: 'Problem 447: Retractions C'
|
|
challengeType: 5
|
|
forumTopicId: 302119
|
|
dashedName: problem-447-retractions-c
|
|
---
|
|
|
|
# --description--
|
|
|
|
For every integer $n > 1$, the family of functions $f_{n, a, b}$ is defined by:
|
|
|
|
$f_{n, a, b}(x) ≡ ax + b\bmod n$ for $a, b, x$ integer and $0 \lt a \lt n$, $0 \le b \lt n$, $0 \le x \lt n$.
|
|
|
|
We will call $f_{n, a, b}$ a retraction if $f_{n, a, b}(f_{n, a, b}(x)) \equiv f_{n, a, b}(x)\bmod n$ for every $0 \le x \lt n$.
|
|
|
|
Let $R(n)$ be the number of retractions for $n$.
|
|
|
|
$F(N) = \displaystyle\sum_{n = 2}^N R(n)$.
|
|
|
|
$F({10}^7) ≡ 638\\,042\\,271\bmod 1\\,000\\,000\\,007$.
|
|
|
|
Find $F({10}^{14})$. Give your answer modulo $1\\,000\\,000\\,007$.
|
|
|
|
# --hints--
|
|
|
|
`retractionsC()` should return `530553372`.
|
|
|
|
```js
|
|
assert.strictEqual(retractionsC(), 530553372);
|
|
```
|
|
|
|
# --seed--
|
|
|
|
## --seed-contents--
|
|
|
|
```js
|
|
function retractionsC() {
|
|
|
|
return true;
|
|
}
|
|
|
|
retractionsC();
|
|
```
|
|
|
|
# --solutions--
|
|
|
|
```js
|
|
// solution required
|
|
```
|