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[数学]割り切った関係

たまには頭の体操を。

ある数を特定の数で割り切れるか、を調べる方法の話です。

例えば、2で割り切れるか、は偶数かどうか見ればよいので下1桁が偶数かどうかみます。
同様に5で割り切れるかも、下1桁が0か5か見ればよいです。

他に有名なのは3で割り切れるかですね。
各桁の数字を足していって、その合計が3で割り切れれば元の数も3で割り切れます。
これも同様に各桁の合計が9で割り切れれば、元の数も9で割り切れます。

さて…これを踏まえて…。
ある数が37で割り切れるか調べる簡単な方法を考えてください。

投稿日時 : 2009年7月22日 12:32

Feedback

# re: [数学]割り切った関係 2009/07/22 13:03 ごう

#手で計算して気づいた。
まず、3桁のぞろ目。
それ以外については調べ方が思いつきません(・ω・;)

# re: [数学]割り切った関係 2009/07/22 13:20 みきぬ

4桁までなら。

・数字を2桁ずつに分ける(7918 → 79 18)
・それぞれについて、37で割った余りを求める(5 18)
・先のほうの数字を11倍して、そこから後のほうの数字を引く(5*11-18 = 37)

出てきた数字なら37の倍数ならおっけー。

# re: [数学]割り切った関係 2009/07/22 13:29 ぱると

37で割ってみる。

多分みきぬさんのやり方より早そうな…(汗

# re: [数学]割り切った関係 2009/07/22 13:42 みきぬ

一般化できたような。

・数字を2桁ずつに分ける(2059235 → 2 05 92 35)
・それぞれについて、37で割った余りを求める(2 05 18 35)
・順序をひっくり返す(35 18 05 2)
・これを足していく。(35-18*11+5*10+2 = -111)
 -3n-2番目の数字はそのまま足す。(n≧1)
 -3n-1番目のは11倍して引く。
 -3n番目のは10倍して足す。

出てきた数字が37の倍数ならおっけー。

# re: [数学]割り切った関係 2009/07/22 13:56 みきぬ

もっと簡単にできたorz

・数字を3桁ずつに分ける(4568427 → 4 568 427)
・それぞれを足した合計が 37 の倍数ならおっけー。

# re: [数学]割り切った関係 2009/07/22 16:03 n

>ある数が37で割り切れるか調べる簡単な方法を考えてください。

「37で割り切れるある数をすべて」ではなく
「ある数が37で割り切れるか」なら割ってみればOKですね。

# re: [数学]割り切った関係 2009/07/22 16:24 よねけん

・ぞろ目はたぶん37の倍数である。(証明してませんが・・・)
・”ある数”に最も近いぞろ目の値から+/-37を何回か繰り返して、”ある数”になればOK

ってのはどうでしょう。

# re: [数学]割り切った関係 2009/07/22 16:26 よねけん

と思ったけど、3桁の場合以外はぞろ目になってませんねorz

# re: [数学]割り切った関係 2009/07/23 9:17 επιστημη

3桁のゾロ目は111の倍数。
111 = 37 x 3 であるから、111の倍数なら37の倍数でもある。
なので3桁ゾロ目は37の倍数。

# re: [数学]割り切った関係 2009/07/23 10:10 みきぬ

昨日の証明。

x を、0以上999999以下の整数とします。
0以上999以下の整数a, bを使ってこう表すことができます。
x = 1000a + b
ここで、999は37の倍数なので、
x = 1000a + b = 999a + a + b = 37 * 27a + a + b
こう書けます。
つまり、a+bが37の倍数であれば、xも37の倍数ということになります。

7桁以上の場合も同様ということで。

# re: [数学]割り切った関係 2009/07/23 12:39 出水

999が37の倍数って事で、3桁ごとに区切って足せばいいですね

最初は11だったけど、意図が見えにくいように37にしました

# re: [数学]割り切った関係 2009/07/23 14:10 みきぬ

11の場合、2桁ごとに区切って足す以外にも求め方がありますね。

奇数桁と偶数桁に分けて、奇数桁の数字は足して、偶数桁の数字は引いていきます。
その合計が、11の倍数であればOKです。

例:81829 → 8-1+8-2+9 = 22 よって11の倍数。

# re: [数学]割り切った関係 2009/07/23 16:19 kira

>a+bが37の倍数であれば、xも37の倍数ということになります

「ある数が37の倍数かどうか調べる」のに、
「違う数が37の倍数かどうか見る」工程いるんだったら
「最初からある数が37の倍数かどうか見ればいいじゃん」
と思う。

# re: [数学]割り切った関係 2009/07/23 17:32 みきぬ

> 「最初からある数が37の倍数かどうか見ればいいじゃん」と思う。

問題文に「さて…これを踏まえて…。」以前の部分がなければ、それでもいいでしょうね。

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