Question

the number of integral points (x,y) which lie in first quadrant but not on coordinate axes and also on straight line 3x+5y = 2007 is ?

Answer 1

Dear student
$$\begin{gathered} ATQ \\ Itmeansx>0andy>0 \\ and3x=2007-5y \\ x=\frac{2007-5y}{3} \\ \Rightarrow x=669-\frac{5y}{3} \\ Sincexisaninteger \\ \Rightarrow \frac{5y}{3}isaninteger. \\ \Rightarrow yisdivisibleby3 \\ and669-\frac{5y}{3}>0 \\ \Rightarrow y<\frac{2007}{5} \\ \Rightarrow y\le 401 \\ No.ofmultiplesofthreefrom1to401 \\ Weseethat3divises399ie\frac{399}{3}=133 \\ Hencethereare133points. \end{gathered}$$

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