2022-01-20 19:30:18 +00:00
|
|
|
---
|
|
|
|
id: 5900f4091000cf542c50ff1c
|
|
|
|
title: '問題 157: ディオファントス方程式を解く'
|
|
|
|
challengeType: 5
|
|
|
|
forumTopicId: 301788
|
|
|
|
dashedName: problem-157-solving-the-diophantine-equation
|
|
|
|
---
|
|
|
|
|
|
|
|
# --description--
|
|
|
|
|
|
|
|
$a$, $b$, $p$, $n$ を正の整数とし、$a ≤ b$ を満たすディオファントス方程式 $\frac{1}{a} + \frac{1}{b} = \frac{p}{{10}^n}$ について考えます。
|
|
|
|
|
|
|
|
$n = 1$ のとき、この方程式は次の 20 個の解を持ちます。
|
|
|
|
|
2022-04-02 08:46:30 +00:00
|
|
|
$$\begin{array}{lllll} \frac{1}{1} + \frac{1}{1} = \frac{20}{10} & \frac{1}{1} + \frac{1}{2} = \frac{15}{10} & \frac{1}{1} + \frac{1}{5} = \frac{12}{10} & \frac{1}{1} + \frac{1}{10} = \frac{11}{10} & \frac{1}{2} + \frac{1}{2} = \frac{10}{10} \\\\
|
|
|
|
\frac{1}{2} + \frac{1}{5} = \frac{7}{10} & \frac{1}{2} + \frac{1}{10} = \frac{6}{10} & \frac{1}{3} + \frac{1}{6} = \frac{5}{10} & \frac{1}{3} + \frac{1}{15} = \frac{4}{10} & \frac{1}{4} + \frac{1}{4} = \frac{5}{10} \\\\
|
|
|
|
\frac{1}{4} + \frac{1}{4} = \frac{5}{10} & \frac{1}{5} + \frac{1}{5} = \frac{4}{10} & \frac{1}{5} + \frac{1}{10} = \frac{3}{10} & \frac{1}{6} + \frac{1}{30} = \frac{2}{10} & \frac{1}{10} + \frac{1}{10} = \frac{2}{10} \\\\
|
|
|
|
\frac{1}{11} + \frac{1}{110} = \frac{1}{10} & \frac{1}{12} + \frac{1}{60} = \frac{1}{10} & \frac{1}{14} + \frac{1}{35} = \frac{1}{10} & \frac{1}{15} + \frac{1}{30} = \frac{1}{10} & \frac{1}{20} + \frac{1}{20} = \frac{1}{10} \end{array}$$
|
2022-01-20 19:30:18 +00:00
|
|
|
|
|
|
|
$1 ≤ n ≤ 9$ のとき、この方程式の解はいくつありますか。
|
|
|
|
|
|
|
|
# --hints--
|
|
|
|
|
|
|
|
`diophantineEquation()` は `53490` を返す必要があります。
|
|
|
|
|
|
|
|
```js
|
|
|
|
assert.strictEqual(diophantineEquation(), 53490);
|
|
|
|
```
|
|
|
|
|
|
|
|
# --seed--
|
|
|
|
|
|
|
|
## --seed-contents--
|
|
|
|
|
|
|
|
```js
|
|
|
|
function diophantineEquation() {
|
|
|
|
|
|
|
|
return true;
|
|
|
|
}
|
|
|
|
|
|
|
|
diophantineEquation();
|
|
|
|
```
|
|
|
|
|
|
|
|
# --solutions--
|
|
|
|
|
|
|
|
```js
|
|
|
|
// solution required
|
|
|
|
```
|