【数学A】例題4.2.14：n進法の位の数（One More）★★★

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% 例題A4.2.14：$n$進法の位の数（One More）★★★
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自然数$N$を6進法と9進法で表すと，それぞれ3桁の数$abc_{(6)}$と$cab_{(9)}$になるとする．このとき$a,b,c$の値を求めよ．また，$N$を10進法で表せ．

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% 例題A4.2.14の解答（One More）★★★
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自然数$N$を6進法，9進法で表すと，それぞれ$abc_{(6)}$と$cab_{(9)}$であるから，
\[
1\leqq a\leqq 5,0\leqq b\leqq 5,1\leqq c\leqq 5
\]

$abc_{(6)},cab_{(9)}$をそれぞれ10進法で表すと，

\begin{align*}
&abc_{(6)}=a\cdot 6^2+b\cdot 6^1+c\cdot 6^0=36a+6b+c\cdots(\mathrm{i}),\\
&cab_{(9)}=c\cdot 9^2+a\cdot 9^1+b\cdot 9^0=81c+9a+b
\end{align*}

これらが等しいから，$36a+6b+c=81c+9a+b$

したがって，$27a=80c-5b$，すなわち，$27a=5(16c-b)\cdots(\mathrm{ii})$

これより，5と27は互いに素であるから，$a$は5の倍数である．

ゆえに，$1\leqq a\leqq 5$より，$a=5$

これを(ii)に代入して整理すると，$16c=b+27\cdots(\mathrm{iii})$

これより，$b+27$は16の倍数である．

したがって，$0\leqq b\leqq 5$より，$27\leqq b+27\leqq 32$であるから，$b+27=32$

ゆえに，$b=5$であり，(iii)より，$1\leqq c\leqq 5$を満たす整数$c$は，$c=2$

よって，$a=5,b=5,c=2$

また，この値を(i)に代入すると，$N=36\cdot 5+6\cdot 5+2=212$

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% 問題A4.2.14：$n$進法の位の数（One More）★★★
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自然数$N$を4進法と7進法で表すと，それぞれ2桁の数$ab_{(4)}$と$ba_{(7)}$になるとする．このとき$a,b$の値を求めよ．また，$N$を10進法で表せ．

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% 問題A4.2.14の解答（One More）★★★
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自然数$N$を4進法，7進法で表すと，それぞれ$ab_{(4)}$と$ba_{(7)}$であるから，
\[
1\leqq a\leqq 3,1\leqq b\leqq 3\cdots(\mathrm{i})
\]

$ab_{(4)},ba_{(7)}$をそれぞれ10進法で表すと，

\begin{align*}
&ab_{(4)}=a\cdot 4^1+b\cdot 4^0=4a+b\cdots(\mathrm{ii}),\\
&ba_{(7)}=b\cdot 7^1+a\cdot 7^0=7b+a
\end{align*}

これらが等しいから，$4a+b=7b+a$

したがって，$a=2b$

よって，(i)において，これを満たす$a,b$を求めると，$a=2,b=1$

また，この値を(ii)に代入すると，$N=4\cdot 2+1=9$

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% 例題A4.2.14の側注：$n$進法の位の数（One More）★★★
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% 側注1
$a,b,c$は6進法の数$abc_{(6)}$の各位の数字であるから，$a,b,c$はそれぞれ$0,1,2,3,4,5$のいずれかの整数である．また，$abc_{(6)}$と$cab_{(9)}$はどちらも3桁の数であるから，最高位になる$a,c$は，$a\neq 0,c\neq 0$であることに注意すること．

% 側注2
$a$の値を定める．

% 側注3
$27\cdot 5=5(16c-b)$を整理すると，(iii)が得られる．

% 側注4
$cab_{(9)}$の式に代入して計算してもよい．

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% 問題A4.2.14の側注：$n$進法の位の数（One More）★★★
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% 側注1
$a,b$は4進法の数$ab_{(4)}$の各位の数字であるから，$a,b$はそれぞれ$0,1,2,3$のいずれかの整数である．また，$ab_{(4)}$と$ba_{(7)}$はどちらも2桁の数であるから，最高位になる$a,b$は，$a\neq 0,b\neq 0$であることに注意すること．

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% 例題A4.2.14の考え方・補足など：$n$進法の位の数（One More）★★★
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% 考え方
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$n$進法で表された数は，10進法で表すと扱いやすくなることがある．ここでは，6進法と9進法で表されているそれぞれの数を，10進法で表すことを考える．$n$進数の各位の数は，最高位以外は$0$以上$n-1$以下の整数となることを利用して，$a,b,c$の値を定める．

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$n$進数の各位の数は，$0$以上$n-1$以下の整数となる．