問題の解答
検索用コード（LaTeX）
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% 例題A4.2.15：$n$進数の利用（One More）★★
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$0,1,2,3$の4種類の数字のみを用いて表される自然数を，小さい方から順に並べると，
\[
1,2,3,10,11,12,13,20,21,22,23,30,31,32,33,100,\ldots
\]
となる．このとき，次の問いに答えよ．
(1) 3121は小さい方から何番目の数であるかを求めよ．
(2) 小さい方から123番目の数を求めよ．
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% 例題A4.2.15の解答（One More）★★
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$0,1,2,3$の4種類の数字で表されているので，この数の列は，4進法で表されている．
(1) $3121_{(4)}$を10進法で表すと，
\[
3121_{(4)}=3\times 4^3+1\times 4^2+2\times 4+1=217
\]
よって，3121は，217番目の数である．
(2) 123を4進法で表すと，
\[
123=1323_{(4)}
\]
よって，123番目の数は，1323である．
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% 問題A4.2.15：$n$進数の利用（One More）★★
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$0,1,2$の3種類の数字のみを用いて表される自然数を，小さい方から順に並べると，
\[
1,2,10,11,12,20,21,22,100,\ldots
\]
となる．このとき，次の問いに答えよ．
(1) 2102は小さい方から何番目の数であるかを求めよ．
(2) 小さい方から87番目の数を求めよ．
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% 問題A4.2.15の解答（One More）★★
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$0,1,2$の3種類の数字で表されているので，この数の列は，3進法で表されている．
(1) $2102_{(3)}$を10進法で表すと，
\[
2102_{(3)}=2\times 3^3+1\times 3^2+0\times 3^1+2=65
\]
よって，2102は，65番目の数である．
(2) 87を3進法で表すと，
\[
87=10020_{(3)}
\]
よって，87番目の数は，10020である．
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