51 lines
970 B
Markdown
51 lines
970 B
Markdown
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---
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id: 5900f52c1000cf542c51003d
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title: 'Problem 446: Retractions B'
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challengeType: 5
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forumTopicId: 302118
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dashedName: problem-446-retractions-b
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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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$F(N) = \displaystyle\sum_{n = 1}^N R(n^4 + 4)$.
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$F(1024) = 77\\,532\\,377\\,300\\,600$.
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Find $F({10}^7)$. Give your answer modulo $1\\,000\\,000\\,007$.
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# --hints--
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`retractionsB()` should return `907803852`.
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```js
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assert.strictEqual(retractionsB(), 907803852);
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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 retractionsB() {
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return true;
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}
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retractionsB();
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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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