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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) = ?
a,b,c∈N5a+6b+7c=70find:max(a)=?
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)
If0∈N⇒maxa=14(b=c=0)If0∉N⇒maxa=10(b=1∧c=2)
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
let6b+7c=d7=6(1)+1⇒1=6(−1)+7(1)⇒d=6(−d)+7(d)⇒b=−d+7t,c=d−6t5a+d=705=1(4)+1⇒1=5(1)+1(−4)⇒70=5(70)+1(−280)⇒a=70+k,d=−280−5k{a=70+kb=280+5k+7tc=−280−5t−6ta>0⇒k>−70b>0⇒7t>−280−5(−70)⇒t>1c>0⇒−6t>280+5(−70)⇒t⩽11hence2⩽t⩽11

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