Largest possible value

Number Theory Level 4

Determine the biggest possible value of k k for which the equation 2005 x + 2007 y = k 2005x + 2007y = k has unique solution in natural numbers.


The answer is 8048070.

Vilakshan Gupta by Vilakshan Gupta , 19, India

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1 solution

Patrick Corn
Oct 9, 2017

If ( x , y ) (x,y) is a solution, then so is ( x − 2007 , y + 2005 ) (x-2007,y+2005) and ( x + 2007 , y − 2005 ) . (x+2007,y-2005). So for ( x , y ) (x,y) to be unique, we must have x − 2007 ≤ 0 x-2007 \le 0 and y − 2005 ≤ 0. y-2005 \le 0. The largest such ( x , y ) (x,y) is ( 2007 , 2005 ) , (2007,2005), which leads to k = 2005 ⋅ 2007 ⋅ 2 = 8048070 . k=2005 \cdot 2007 \cdot 2 = \fbox{8048070}.

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