52 lines
1.2 KiB
Markdown
52 lines
1.2 KiB
Markdown
---
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id: 5900f4ff1000cf542c510011
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title: 'Problem 402: Integer-valued polynomials'
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challengeType: 5
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forumTopicId: 302070
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dashedName: problem-402-integer-valued-polynomials
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---
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# --description--
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It can be shown that the polynomial $n^4 + 4n^3 + 2n^2 + 5n$ is a multiple of 6 for every integer $n$. It can also be shown that 6 is the largest integer satisfying this property.
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Define $M(a, b, c)$ as the maximum $m$ such that $n^4 + an^3 + bn^2 + cn$ is a multiple of $m$ for all integers $n$. For example, $M(4, 2, 5) = 6$.
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Also, define $S(N)$ as the sum of $M(a, b, c)$ for all $0 < a, b, c ≤ N$.
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We can verify that $S(10) = 1\\,972$ and $S(10\\,000) = 2\\,024\\,258\\,331\\,114$.
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Let $F_k$ be the Fibonacci sequence:
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- $F_0 = 0$, $F_1 = 1$ and
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- $F_k = F_{k - 1} + F_{k - 2}$ for $k ≥ 2$.
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Find the last 9 digits of $\sum S(F_k)$ for $2 ≤ k ≤ 1\\,234\\,567\\,890\\,123$.
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# --hints--
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`integerValuedPolynomials()` should return `356019862`.
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```js
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assert.strictEqual(integerValuedPolynomials(), 356019862);
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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 integerValuedPolynomials() {
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return true;
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}
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integerValuedPolynomials();
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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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