Integers

To solve for the smallest positive integer such that , we begin by determining the value of the expression .

First, rationalize the denominator by multiplying the numerator and the denominator by the conjugate of the denominator:

.

Calculate the numerator:

(since ).

Calculate the denominator:

.

Thus, .

Now, we need . The powers of cycle every 4 terms as follows:

To achieve , must be a multiple of 4. The smallest positive integer satisfying this is .