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If x and y are positive integers such that 5x+3y=100, what is the greatest possible value of xy?

Feb 25, 2019

#1
+22499
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If x and y are positive integers such that 5x+3y=100,

what is the greatest possible value of xy?

$$\begin{array}{|rcll|} \hline 5x+3y &=& 100 \\ 3y &=& 100 - 5x \quad | \quad (3 < 5)! \\ y &=& \dfrac{100 - 5x} {3} \\ y &=& \dfrac{99+1-5x-x+x} {3} \\ y &=& \dfrac{99-6x+1+x} {3} \\ y &=& 33-2x+ \underbrace{\dfrac{1+x} {3}}_{=a} \\\\ a &=& \dfrac{1+x} {3}\\ 3a &=& 1+x \\ \mathbf{x} &\mathbf{=}& \mathbf{3a-1} \quad | \quad a \in \mathbb{Z} \\\\ y &=& 33-2x+ \dfrac{1+x} {3} \quad | \quad x = 3a-1 \\ y &=& 33-2(3a-1)+ \dfrac{1+3a-1} {3} \\ y &=& 33-6a+2+ \dfrac{3a} {3} \\ y &=& 33-6a+2+a \\ y &=& 35-5a \\ \mathbf{y} &\mathbf{=}& \mathbf{5(7-a)} \quad | \quad a \in \mathbb{Z} \\ \hline \end{array}$$

$$\begin{array}{|r|r|r|r|} \hline a & x=3a-1 & y=5(7-a) & xy & \text{max} \\ \hline 1 & 2 & 30 & 60 & \\ \hline 2 & 5 & 25 & 125 & \\ \hline 3 & 8 & 20 & 160 & \\ \hline 4 & 11 & 15 & 165 & \checkmark \\ \hline 5 & 14 & 10 & 140 & \\ \hline 6 & 17 & 5 & 85 & \\ \hline 7 & 20 & 0 & 0 & \\ \hline \end{array}$$

The greatest possible value of xy is 165.

Feb 25, 2019