---
id: 5900f5241000cf542c510037
title: 'Problem 440: GCD and Tiling'
challengeType: 5
forumTopicId: 302112
dashedName: problem-440-gcd-and-tiling
---

# --description--

We want to tile a board of length $n$ and height 1 completely, with either 1 × 2 blocks or 1 × 1 blocks with a single decimal digit on top:

<img class="img-responsive center-block" alt="ten blocks 1x1 with single decimal digit on top, and 1x2 block" src="https://cdn.freecodecamp.org/curriculum/project-euler/gcd-and-tiling-1.png" style="background-color: white; padding: 10px;">

For example, here are some of the ways to tile a board of length $n = 8$:

<img class="img-responsive center-block" alt="examples of ways to tile a board of length n = 8" src="https://cdn.freecodecamp.org/curriculum/project-euler/gcd-and-tiling-2.png" style="background-color: white; padding: 10px;">

Let $T(n)$ be the number of ways to tile a board of length $n$ as described above.

For example, $T(1) = 10$ and $T(2) = 101$.

Let $S(L)$ be the triple sum $\sum_{a, b, c} gcd(T(c^a), T(c^b))$ for $1 ≤ a, b, c ≤ L$.

For example:

$$\begin{align}
  & S(2) = 10\\,444 \\\\
  & S(3) = 1\\,292\\,115\\,238\\,446\\,807\\,016\\,106\\,539\\,989 \\\\
  & S(4)\bmod 987\\,898\\,789 = 670\\,616\\,280.
\end{align}$$

Find $S(2000)\bmod 987\\,898\\,789$.

# --hints--

`gcdAndTiling()` should return `970746056`.

```js
assert.strictEqual(gcdAndTiling(), 970746056);
```

# --seed--

## --seed-contents--

```js
function gcdAndTiling() {

  return true;
}

gcdAndTiling();
```

# --solutions--

```js
// solution required
```