Combinations!?
Let S={(a, b) : a, b Z, } .
The number of elements (x, y) in S such that 3x+4y+5 is divisible by 19.
Note :This question is a part of set KVPY 2014 SB
The answer is 19.
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1 solution
3x+4y+5=19w now min(3x+4y+5)=5 and max (3x+4y+5)=131 and hence 5<=19w<=131 now we can easily check what all values w can take w=1,2,3,4,5,6 now : 1)w=1 =>1 solution 2)w=2 =>3 solution 3)w=3 =>5 solutions 4)w=4 =>5 solutions 5)w=5 =>4 solutions 6)w=6 => 1solution
total solutions=1+3+5+5+4+1=19