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 ?

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Solution

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$A T Q I t m e a n s x > 0 a n d y > 0 a n d 3 x = 2007 - 5 y x = \frac{2007 - 5 y}{3} \Rightarrow x = 669 - \frac{5 y}{3} S i n c e x i s a n i n t e g e r \Rightarrow \frac{5 y}{3} i s a n i n t e g e r . \Rightarrow y i s d i v i s i b l e b y 3 a n d 669 - \frac{5 y}{3} > 0 \Rightarrow y < \frac{2007}{5} \Rightarrow y \leq 401 N o . o f m u l t i p l e s o f t h r e e f r o m 1 t o 401 W e s e e t h a t 3 d i v i s e s 399 i e \frac{399}{3} = 133 H e n c e t h e r e a r e 133 p o i n t s .$
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