To find: The number of lattice points in the 1st quadrant, lying on straight line: $3x+5y = 283.$
-I tried this question a lot many times. The long substitution method becomes tedious. Can you please show a simpler and faster way of solving this question? I appreciate the help. Thanks.
Try $ 3(an + b) + 5 (cn + d) = 283 $
Then $ 3an + 5cn = 0 $ and $ 3b + 5d = 283 $
lets try to solve the last equation by noticing that if $b = 1$ then $5d = 280$
, then $d = 56$ .
Now $3na + 5nc = 0$ , $n(3a + 5b) = 0 $, try $a = 5$ , $c= -3$.
Putting it all together: $3*(5n+1) + 5*(-3n + 56) = 283 $ for all n.
Just a little more work to do if you want the answer in the first quadrant.