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a-b-c-N-5a-6b-7c-70-find-max-a-

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Question Number 212085 by hardmath last updated on 28/Sep/24
a,b,c ∈ N 5a + 6b + 7c = 70 find: max(a) = ?
$$\mathrm{a},\mathrm{b},\mathrm{c}\:\in\:\mathbb{N} \\ $$$$\mathrm{5a}\:+\:\mathrm{6b}\:+\:\mathrm{7c}\:=\:\mathrm{70} \\ $$$$\mathrm{find}:\:\:\mathrm{max}\left(\mathrm{a}\right)\:=\:? \\ $$
Commented by Frix last updated on 28/Sep/24
If 0∈N ⇒ max a =14 (b=c=0) If 0∉N ⇒ max a =10 (b=1∧c=2)
$$\mathrm{If}\:\mathrm{0}\in\mathbb{N}\:\Rightarrow\:\mathrm{max}\:{a}\:=\mathrm{14}\:\:\:\:\:\left({b}={c}=\mathrm{0}\right) \\ $$$$\mathrm{If}\:\mathrm{0}\notin\mathbb{N}\:\Rightarrow\:\mathrm{max}\:{a}\:=\mathrm{10}\:\:\:\:\:\left({b}=\mathrm{1}\wedge{c}=\mathrm{2}\right) \\ $$
Answered by alcohol last updated on 05/Oct/24
let 6b + 7c = d 7 = 6(1) + 1 ⇒ 1 = 6(−1) + 7(1) ⇒ d = 6(−d) + 7(d) ⇒ b = −d + 7t, c = d − 6t 5a + d = 70 5 = 1(4) + 1 ⇒ 1 = 5(1) + 1(−4) ⇒ 70 = 5(70) + 1(−280) ⇒ a = 70 + k, d = −280 − 5k { ((a = 70 + k)),((b = 280 + 5k + 7t)),((c = −280 − 5t −6t)) :} a>0 ⇒ k > −70 b >0 ⇒7t > −280 −5(−70) ⇒ t > 1 c >0 ⇒ −6t > 280 +5(−70) ⇒ t ≤ 11 hence 2 ≤ t ≤ 11
$${let}\:\mathrm{6}{b}\:+\:\mathrm{7}{c}\:=\:{d} \\ $$$$\mathrm{7}\:=\:\mathrm{6}\left(\mathrm{1}\right)\:+\:\mathrm{1} \\ $$$$\Rightarrow\:\mathrm{1}\:=\:\mathrm{6}\left(−\mathrm{1}\right)\:+\:\mathrm{7}\left(\mathrm{1}\right) \\ $$$$\Rightarrow\:{d}\:=\:\mathrm{6}\left(−{d}\right)\:+\:\mathrm{7}\left({d}\right) \\ $$$$\Rightarrow\:{b}\:=\:−{d}\:+\:\mathrm{7}{t},\:{c}\:=\:{d}\:−\:\mathrm{6}{t} \\ $$$$\mathrm{5}{a}\:+\:{d}\:=\:\mathrm{70} \\ $$$$\mathrm{5}\:=\:\mathrm{1}\left(\mathrm{4}\right)\:+\:\mathrm{1} \\ $$$$\Rightarrow\:\mathrm{1}\:=\:\mathrm{5}\left(\mathrm{1}\right)\:+\:\mathrm{1}\left(−\mathrm{4}\right) \\ $$$$\Rightarrow\:\mathrm{70}\:=\:\mathrm{5}\left(\mathrm{70}\right)\:+\:\mathrm{1}\left(−\mathrm{280}\right) \\ $$$$\Rightarrow\:{a}\:=\:\mathrm{70}\:+\:{k},\:{d}\:=\:−\mathrm{280}\:−\:\mathrm{5}{k} \\ $$$$\begin{cases}{{a}\:=\:\mathrm{70}\:+\:{k}}\\{{b}\:=\:\mathrm{280}\:+\:\mathrm{5}{k}\:+\:\mathrm{7}{t}}\\{{c}\:=\:−\mathrm{280}\:−\:\mathrm{5}{t}\:−\mathrm{6}{t}}\end{cases} \\ $$$${a}>\mathrm{0}\:\Rightarrow\:{k}\:>\:−\mathrm{70} \\ $$$${b}\:>\mathrm{0}\:\Rightarrow\mathrm{7}{t}\:>\:−\mathrm{280}\:−\mathrm{5}\left(−\mathrm{70}\right) \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\Rightarrow\:{t}\:>\:\mathrm{1} \\ $$$${c}\:>\mathrm{0}\:\Rightarrow\:−\mathrm{6}{t}\:>\:\mathrm{280}\:+\mathrm{5}\left(−\mathrm{70}\right) \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\Rightarrow\:{t}\:\leqslant\:\mathrm{11} \\ $$$${hence}\:\mathrm{2}\:\leqslant\:{t}\:\leqslant\:\mathrm{11} \\ $$$$ \\ $$

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