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# INMO 2013 Question No. 3 Solution 3     Let such that . Show that the equation has no integer solution.

Sketch of the Solution:

Claim 1: There cannot be a negative integer solution. Suppose other wise. If possible x= -k (k positive) be a solution.

Then we have . Clearly this is impossible as and .

Claim 2: 0 is not a solution (why?)

Claim 3: There cannot be a positive integer solution. Suppose other wise. If possible x=k (k positive) be a solution.

Then we have This implies that the right hand side is divisible by k which again implies that d is divisible by k (why?).
Let d=d'k
Now .
Thus .
Hence the equality is impossible.

3     Let such that . Show that the equation has no integer solution.

Sketch of the Solution:

Claim 1: There cannot be a negative integer solution. Suppose other wise. If possible x= -k (k positive) be a solution.

Then we have . Clearly this is impossible as and .

Claim 2: 0 is not a solution (why?)

Claim 3: There cannot be a positive integer solution. Suppose other wise. If possible x=k (k positive) be a solution.

Then we have This implies that the right hand side is divisible by k which again implies that d is divisible by k (why?).
Let d=d'k
Now .
Thus .
Hence the equality is impossible.

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