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Explore the Back-Story3 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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