2017年12月29日金曜日

数学 - Python - JavaScript - 代数学 - 整数 - 同値関係、合同式(法の約数を法とする)

1. d を m の任意の約数とすれば、ある整数 k が存在して、

$m=dk\left(k\in \text{ℤ}\right)$

とおくことができる。

よって、

$\begin{array}{}a-b=dks\left(s\in \text{ℤ}\right)\\ a-b=\left(ks\right)d\end{array}$

ゆえに、

$a\equiv b\left(mod\left(d\right)\right)$

コード(Emacs)

Python 3

#!/usr/bin/env python3
from sympy import pprint, gcd
import random

def mod(a, b, m):
return (a - b) % m == 0

for _ in range(10):
a = random.randrange(-100, 100)
b = random.randrange(-100, 100)
m = random.randrange(1, 20)
b += ((a % m) - (b % m))
for d in range(1, m + 1):
if m % d == 0:
break
print(f'a = {a}, b = {b}, m = {m}, d = {d}')
for t in [mod(a, b, m), mod(a, b, d)]:
pprint(t)
print()


$./sample5.py a = -98, b = -93, m = 5, d = 1 True True a = -14, b = 25, m = 13, d = 1 True True a = 13, b = -55, m = 4, d = 1 True True a = 13, b = -52, m = 1, d = 1 True True a = -12, b = 20, m = 2, d = 1 True True a = 79, b = 24, m = 5, d = 1 True True a = 79, b = -41, m = 10, d = 1 True True a = 98, b = -72, m = 10, d = 1 True True a = 8, b = 62, m = 9, d = 1 True True a = -80, b = 80, m = 10, d = 1 True True$


HTML5

<pre id="output0"></pre>

<input id="a0" type="number" step="1" value="12">
≡
<input id="b0" type="number" step="1" value="22">
(mod <input id="m0" type="number" step="1" value="10">)
<br>
d = <input id="d0" type="number" min="1" step="1" value="2">
<button id="run0">run</button>
<button id="clear0">clear</button>
<script src="sample5.js"></script>


JavaScript

let pre0 = document.querySelector('#output0'),
run0 = document.querySelector('#run0'),
clear0 = document.querySelector('#clear0'),
input_a0 = document.querySelector('#a0'),
input_b0 = document.querySelector('#b0'),
input_m0 = document.querySelector('#m0'),
input_d0 = document.querySelector('#d0'),
inputs = [input_a0, input_b0, input_m0, input_d0],
range = (n) => {
let result = [];

for (let i = 0; i < n; i += 1) {
result.push(i);
}

return result;
},
p = (text) => pre0.textContent += text + '\n',
clear = () => pre0.textContent = '',
mod = (a, b, m) => (a - b) % m === 0,
output = () => {
let a = parseInt(input_a0.value, 10),
b = parseInt(input_b0.value, 10),
m = parseInt(input_m0.value, 10),
d = parseInt(input_d0.value, 10);

if ((a - b) % m == 0) {
if (m % d == 0) {
p(mod(a, b, m));
p(mod(a, b, d));
} else {
p(${d}は${m}の約数ではない);
}
} else {
p(${a}と${b}は\${m}を法として合同ではない);
}
};

run0.onclick = output;
clear0.onclick = clear;
inputs.forEach((input) => input.onchange = output);
output();


(mod )

d =