51 lines
1.0 KiB
Markdown
51 lines
1.0 KiB
Markdown
---
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id: 5900f52a1000cf542c51003c
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title: 'Problem 445: Retractions A'
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challengeType: 5
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forumTopicId: 302117
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dashedName: problem-445-retractions-a
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---
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# --description--
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For every integer $n > 1$, the family of functions $f_{n, a, b}$ is defined by:
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$f_{n, a, b}(x) ≡ ax + b\bmod n$ for $a, b, x$ integer and $0 \lt a \lt n$, $0 \le b \lt n$, $0 \le x \lt n$.
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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 $0 \le x \lt n$.
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Let $R(n)$ be the number of retractions for $n$.
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You are given that
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$$\sum_{k = 1}^{99\\,999} R(\displaystyle\binom{100\\,000}{k}) \equiv 628\\,701\\,600\bmod 1\\,000\\,000\\,007$$
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Find $$\sum_{k = 1}^{9\\,999\\,999} R(\displaystyle\binom{10\\,000\\,000}{k})$$ Give your answer modulo $1\\,000\\,000\\,007$.
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# --hints--
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`retractionsA()` should return `659104042`.
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```js
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assert.strictEqual(retractionsA(), 659104042);
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```
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# --seed--
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## --seed-contents--
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```js
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function retractionsA() {
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return true;
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}
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retractionsA();
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```
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# --solutions--
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```js
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// solution required
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```
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