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Question-207673




Question Number 207673 by efronzo1 last updated on 22/May/24
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Answered by A5T last updated on 22/May/24
abcde=a+b+c+d+e≤5e⇒abcd≤5  abcd=1⇒a=b=c=d=1  ⇒a+b+c+d+e=4+e=e                X  abcd=2⇒a=b=c=1;d=2  ⇒a+b+c+d+e=5+e=2e⇒e=5  abcd=3⇒a=b=c=1;d=3  ⇒a+b+c+d+e=6+e=3e⇒e=3  abcd=4⇒(i): a=b=c=1;d=4  ⇒a+b+c+d+e=7+e=4e            X  (ii):a=b=1;c=d=2  ⇒a+b+c+d+e=6+e=4e⇒e=2  abcd=5⇒a=b=c=1;d=5  ⇒a+b+c+d+e=8+e=5e⇒e=2 (→←)  ⇒maximum value of e=5
$${abcde}={a}+{b}+{c}+{d}+{e}\leqslant\mathrm{5}{e}\Rightarrow{abcd}\leqslant\mathrm{5} \\ $$$${abcd}=\mathrm{1}\Rightarrow{a}={b}={c}={d}=\mathrm{1} \\ $$$$\Rightarrow{a}+{b}+{c}+{d}+{e}=\mathrm{4}+{e}={e}\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:{X} \\ $$$${abcd}=\mathrm{2}\Rightarrow{a}={b}={c}=\mathrm{1};{d}=\mathrm{2} \\ $$$$\Rightarrow{a}+{b}+{c}+{d}+{e}=\mathrm{5}+{e}=\mathrm{2}{e}\Rightarrow{e}=\mathrm{5} \\ $$$${abcd}=\mathrm{3}\Rightarrow{a}={b}={c}=\mathrm{1};{d}=\mathrm{3} \\ $$$$\Rightarrow{a}+{b}+{c}+{d}+{e}=\mathrm{6}+{e}=\mathrm{3}{e}\Rightarrow{e}=\mathrm{3} \\ $$$${abcd}=\mathrm{4}\Rightarrow\left({i}\right):\:{a}={b}={c}=\mathrm{1};{d}=\mathrm{4} \\ $$$$\Rightarrow{a}+{b}+{c}+{d}+{e}=\mathrm{7}+{e}=\mathrm{4}{e}\:\:\:\:\:\:\:\:\:\:\:\:{X} \\ $$$$\left({ii}\right):{a}={b}=\mathrm{1};{c}={d}=\mathrm{2} \\ $$$$\Rightarrow{a}+{b}+{c}+{d}+{e}=\mathrm{6}+{e}=\mathrm{4}{e}\Rightarrow{e}=\mathrm{2} \\ $$$${abcd}=\mathrm{5}\Rightarrow{a}={b}={c}=\mathrm{1};{d}=\mathrm{5} \\ $$$$\Rightarrow{a}+{b}+{c}+{d}+{e}=\mathrm{8}+{e}=\mathrm{5}{e}\Rightarrow{e}=\mathrm{2}\:\left(\rightarrow\leftarrow\right) \\ $$$$\Rightarrow{maximum}\:{value}\:{of}\:{e}=\mathrm{5} \\ $$

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