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FIND ALL PAIRS {x,y}of integers such that xy+ x 3 +y 3 /3= 2007

FIND ALL PAIRS  {x,y}of integers such that
xy+   x3+y3/3= 2007

Grade:9

1 Answers

NIKHIL
22 Points
6 years ago
rewrite the equation as  x3+y3-1+3xy= 6020
factoring both sides we get (x+y-1)(x2+y2+1+x+y-xy) 22.5..7.43
since x2+y2+1+x+y-xy= (x+y-1)2+3x+3y-3xy. now taking the above equality mode 3, we deduce that
x+y-1=2(mod 3)
also x+y-1 is less than x2+y2+1+x+y-xy.so the possible values for x+y-1 are 2,5,14,20 and 35
now finally by inspection we obtained that  x+y-1=20 and finally {x,y} = {18,3}or {3,18}

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