【数学A】例題4.1.4：素因数の個数（One More）★★

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% 例題A4.1.4：素因数の個数（One More）★★
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(1) $13!$が$2^k$で割り切れるとき，自然数$k$の最大値を求めよ．

(2) $60!$は，末尾に0が何個連続して並ぶか．

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% 例題A4.1.4の解答（One More）★★
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(1) 1から13までの自然数について，

2の倍数は6個，$2^2$の倍数は3個，$2^3$の倍数は1個

したがって，$13!$に含まれる因数2の個数は，

\[
6+3+1=10(\text{個})
\]

よって，求める自然数$k$の最大値は，$k=10$

(2) 求める0の個数は$60!$に含まれる因数10の個数に等しい．また，$10=2\cdot 5$であり，$60!$に含まれる因数5の個数が因数2の個数より少ないので，因数10の個数は因数5の個数に等しい．

1から60までの自然数について，

5の倍数は12個，$5^2$の倍数は2個

したがって，$60!$に含まれる因数5の個数は，

\[
12+2=14(\text{個})
\]

よって，求める0の個数は，14個

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% 問題A4.1.4：素因数の個数（One More）★★
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(1) $15!$が$2^k$で割り切れるとき，自然数$k$の最大値を求めよ．

(2) $50!$は，末尾に0が何個連続して並ぶか．

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% 問題A4.1.4の解答（One More）★★
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(1) 1から15までの自然数について，

2の倍数は7個，$2^2$の倍数は3個，$2^3$の倍数は1個

したがって，$15!$に含まれる因数2の個数は，$7+3+1=11(\text{個})$

よって，求める自然数$k$の最大値は，$k=11$

(2) 求める0の個数は$50!$に含まれる因数10の個数に等しい．また，$10=2\cdot 5$であり，$50!$に含まれる因数5の個数が因数2の個数より少ないので，因数10の個数は因数5の個数に等しい．

1から50までの自然数について，

5の倍数は10個，$5^2$の倍数は2個

したがって，$50!$に含まれる因数5の個数は，$10+2=12(\text{個})$

よって，求める0の個数は，12個

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% 例題A4.1.4の側注：素因数の個数（One More）★★
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% 側注1
$2,4,8$の倍数の個数をそれぞれ求める．

\begin{align*}
&13=2\times 6+1,\\
&13=4\times 3+1,\\
&13=8\times 1+5
\end{align*}

一般に，1から$n$までの整数のうち，$k$の倍数の個数は，$n$を$k$で割った商に等しい．

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1から60までの自然数について，2の倍数は30個，5の倍数は12個である．

% 側注3
$5,25$の倍数の個数をそれぞれ求める．

\[
60=5\times 12,
\]

\[
60=25\times 2+10
\]

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% 問題A4.1.4の側注：素因数の個数（One More）★★
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% 側注1
1から50までの自然数について，2の倍数は25個，5の倍数は10個である．

% 側注2
$5,25$の倍数の個数をそれぞれ求める．

\[
50=5\times 10,50=25\times 2
\]

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% 例題A4.1.4の考え方・補足など：素因数の個数（One More）★★
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% 考え方
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(1) $13!\div 2^k=\frac{13\cdot 12\cdot 11\cdot\cdots\cdot 3\cdot 2\cdot 1}{2^k}$であるから，$2^k$で割り切れるとき，自然数$k$の最大値は$13!$が2を因数として何個含むかを考えればよい．

(2) 一の位から連続して並ぶ0の個数は，$60!$に含まれる因数10の個数に等しい．$10=2\cdot 5$であり，$60!$には因数2の方が因数5より多く含まれることを利用する（末尾に並ぶ0の個数は，素因数5の個数と一致する）．

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% 余談
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$n!$に含まれる素因数$p$の個数は，$n$以下の自然数のうち$p,p^2,p^3,\ldots$の倍数となる数の個数の和を考えることで求めることができる．なお，(1)において，$13!$に含まれる素因数2の個数は，次のような表を用いると，因数2の個数が10個となることがわかりやすく理解できる．

\begin{tabular}{|l||c|c|c|c|c|c|c|c|c|c|c|c|c|}
\hline&1&2&3&4&5&6&7&8&9&10&11&12&13\\
\hline 2の倍数&&$\bigcirc$&&$\bigcirc$&&$\bigcirc$&&$\bigcirc$&&$\bigcirc$&&$\bigcirc$&\\
\hline$2^2$の倍数&&&&$\bigcirc$&&&&$\bigcirc$&&&&$\bigcirc$&\\
\hline$2^3$の倍数&&&&&&&&$\bigcirc$&&&&&\\
\hline
\end{tabular}