How many ordered pairs of positive integers (x,y) satisfy the equation x√y+y√x+√2006xy−√2006x−√2006y−2006=0? (correct answer + 3, wrong answer 0)
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Solution
x√y+y√x+√2006xy−√2006x−√2006y−2006=0 ⇒(√x+√y+√2006)(√xy−√2006)=0 ⇒(√xy−√2006)=0(∵√x+√y+√2006>0) ⇒xy=2006=17×59×2 Thus the solutions are (1,2006),(2006,1),(2,1003),(1003,2),(34,59),(59,34),(17,118),(118,17)