【数学A】例題4.1.6：最大公約数・最小公倍数2（One More）★★

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% 例題A4.1.6：最大公約数・最小公倍数2（One More）★★
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次の条件を満たす2つの自然数$a,b$の組をすべて求めよ．ただし，$a<b$とする．

(1) 和が150，最大公約数が10

(2) 積が180，最小公倍数が60

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% 例題A4.1.6の解答（One More）★★
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(1) 最大公約数が10であるから，2つの自然数$a,b$は$a=10a^{\prime},b=10b^{\prime}$とおける．ただし，$a^{\prime},b^{\prime}$は互いに素な自然数であり，$a<b$より，$a^{\prime}<b^{\prime}$

和が150であるから，$10a^{\prime}+10b^{\prime}=150$

したがって，$a^{\prime}+b^{\prime}=15$

これを満たす，互いに素な自然数$a^{\prime},b^{\prime}$の組は，
\[
(a^{\prime},b^{\prime})=(1,14),(2,13),(4,11),(7,8)
\]

よって，

\[
(a,b)=(10,140),(20,130),(40,110),(70,80)
\]

(2) 最大公約数を$g$とすると，積が180，最小公倍数が60であるから，$180=g\cdot 60$

したがって，$g=3$であるから，$a=3a^{\prime},b=3b^{\prime}$とおける．ただし，$a^{\prime},b^{{\prime}}$は互いに素な自然数であり，$a<b$より，$a^{\prime}<b^{\prime}$

$60=3a^{\prime}b^{\prime}$が成り立つから，$a^{\prime}b^{\prime}=20$

これを満たす，互いに素な自然数$a^{\prime},b^{\prime}$の組は，
\[
(a^{\prime},b^{\prime})=(1,20),(4,5)
\]

よって，
\[
(a,b)=(3,60),(12,15)
\]

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% 問題A4.1.6：最大公約数・最小公倍数2（One More）★★
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次の条件を満たす2つの自然数$a,b$の組をすべて求めよ．ただし，$a<b$とする．

(1) 和が180，最大公約数が15

(2) 積が400，最小公倍数が80

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% 問題A4.1.6の解答（One More）★★
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(1) 最大公約数が15であるから，2つの自然数$a,b$は$a=15a^{\prime},b=15b^{\prime}$とおける．ただし，$a^{\prime},b^{\prime}$は互いに素な自然数であり，$a<b$より，$a^{\prime}<b^{\prime}$

和が180であるから，$15a^{\prime}+15b^{\prime}=180$

したがって，$a^{\prime}+b^{\prime}=12$

これを満たす，互いに素な自然数$a^{\prime},b^{\prime}$の組は，
\[
(a^{\prime},b^{\prime})=(1,11),(5,7)
\]

よって，$(a,b)=(15,165),(75,105)$

(2) 最大公約数を$g$とすると，積が400，最小公倍数が80であるから，$400=g\cdot 80$

したがって，$g=5$であるから，$a=5a^{\prime},b=5b^{\prime}$とおける．ただし，$a^{\prime},b^{{\prime}}$は互いに素な自然数であり，$a<b$より，$a^{\prime}<b^{\prime}$

$80=5a^{\prime}b^{\prime}$が成り立つから，$a^{\prime}b^{\prime}=16$

これを満たす，互いに素な自然数$a^{\prime},b^{\prime}$の組は，
\[
(a^{\prime},b^{\prime})=(1,16)
\]

よって，$(a,b)=(5,80)$

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% 例題A4.1.6の側注：最大公約数・最小公倍数2（One More）★★
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% 側注1
$a=ga^\prime,b=gb^\prime$

% 側注2
「$a,b$が互いに素$\Longleftrightarrow a,b$の最大公約数が1」が成り立つ．これより，例えば，$(a^\prime,b^\prime)=(3,12)$は，$a^\prime$と$b^\prime$が互いに素ではないので不適である（最大公約数が1ではない）．

% 側注3
$a=ga^\prime,b=gb^\prime$

% 側注4
$l=a^\prime b^\prime g$

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% 問題A4.1.6の側注：最大公約数・最小公倍数2（One More）★★
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% 側注1
$a=ga^\prime,b=gb^\prime$

% 側注2
「$a,b$が互いに素$\Longleftrightarrow a,b$の最大公約数が1」が成り立つ．これより，例えば，$(a^\prime,b^\prime)=(2,10)$は，$a^\prime$と$b^\prime$が互いに素ではないので不適である（最大公約数が1ではない）．

% 側注3
$a=ga^\prime,b=gb^\prime$

% 側注4
$l=a^\prime b^\prime g$

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% 例題A4.1.6の考え方・補足など：最大公約数・最小公倍数2（One More）★★
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% 考え方
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2つの自然数$a,b$の最大公約数を$g$，最小公倍数を$l$とし，$a=ga^\prime$，$b=gb^\prime$（$a^\prime,b^\prime$は互いに素な自然数）とおき，$l=a^\prime b^\prime g,ab=gl$が成り立つことを利用する．また，$a^\prime,b^\prime$は互いに素であることから，$a^\prime,b^\prime$の値を定める．

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% One Point
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$a,b$の最大公約数が$g$のとき，$a=a^{\prime}g,b=b^{\prime}g$（$a^{\prime}$と$b^{\prime}$は互いに素な自然数）とおく．