---
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
```