49 lines
1.0 KiB
Markdown
49 lines
1.0 KiB
Markdown
|
---
|
||
|
|
id: 5900f4ff1000cf542c510011
|
||
|
|
title: 'Problem 402: Integer-valued polynomials'
|
||
|
|
challengeType: 5
|
||
|
|
forumTopicId: 302070
|
||
|
|
dashedName: problem-402-integer-valued-polynomials
|
||
|
|
---
|
||
|
|
|
||
|
|
# --description--
|
||
|
|
|
||
|
|
It can be shown that the polynomial n4 + 4n3 + 2n2 + 5n is a multiple of 6 for every integer n. It can also be shown that 6 is the largest integer satisfying this property.
|
||
|
|
|
||
|
|
Define M(a, b, c) as the maximum m such that n4 + an3 + bn2 + cn is a multiple of m for all integers n. For example, M(4, 2, 5) = 6.
|
||
|
|
|
||
|
|
Also, define S(N) as the sum of M(a, b, c) for all 0 < a, b, c ≤ N.
|
||
|
|
|
||
|
|
We can verify that S(10) = 1972 and S(10000) = 2024258331114.
|
||
|
|
|
||
|
|
Let Fk be the Fibonacci sequence: F0 = 0, F1 = 1 and Fk = Fk-1 + Fk-2 for k ≥ 2.
|
||
|
|
|
||
|
|
Find the last 9 digits of Σ S(Fk) for 2 ≤ k ≤ 1234567890123.
|
||
|
|
|
||
|
|
# --hints--
|
||
|
|
|
||
|
|
`euler402()` should return 356019862.
|
||
|
|
|
||
|
|
```js
|
||
|
|
assert.strictEqual(euler402(), 356019862);
|
||
|
|
```
|
||
|
|
|
||
|
|
# --seed--
|
||
|
|
|
||
|
|
## --seed-contents--
|
||
|
|
|
||
|
|
```js
|
||
|
|
function euler402() {
|
||
|
|
|
||
|
|
return true;
|
||
|
|
}
|
||
|
|
|
||
|
|
euler402();
|
||
|
|
```
|
||
|
|
|
||
|
|
# --solutions--
|
||
|
|
|
||
|
|
```js
|
||
|
|
// solution required
|
||
|
|
```
|