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Question Number 212085 by hardmath last updated on 28/Sep/24

$$\mathrm{a},\mathrm{b},\mathrm{c}\in \mathbb{N}$$$$5\mathrm{a}+6\mathrm{b}+7\mathrm{c}=70$$$$\mathrm{find}:\max \left(\mathrm{a}\right)=?$$

Commented by Frix last updated on 28/Sep/24

$$\mathrm{If}0\in \mathbb{N}\Rightarrow \max a=14(b=c=0)$$$$\mathrm{If}0\notin \mathbb{N}\Rightarrow \max a=10(b=1\wedge c=2)$$

Answered by alcohol last updated on 05/Oct/24

$$let6b+7c=d$$$$7=6\left(1\right)+1$$$$\Rightarrow 1=6(-1)+7\left(1\right)$$$$\Rightarrow d=6(-d)+7\left(d\right)$$$$\Rightarrow b=-d+7t,c=d-6t$$$$5a+d=70$$$$5=1\left(4\right)+1$$$$\Rightarrow 1=5\left(1\right)+1(-4)$$$$\Rightarrow 70=5\left(70\right)+1(-280)$$$$\Rightarrow a=70+k,d=-280-5k$$$$\{\begin{array}{l}a=70+k\\ b=280+5k+7t\\ c=-280-5t-6t\end{array}$$$$a>0\Rightarrow k>-70$$$$b>0\Rightarrow 7t>-280-5(-70)$$$$\Rightarrow t>1$$$$c>0\Rightarrow -6t>280+5(-70)$$$$\Rightarrow t⩽11$$$$hence2⩽t⩽11$$$$$$

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