Find the smallest positive integer N \neq 23 such that the fraction \frac {N-23} { 7N+6 } is not in simplest terms.

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To find the smallest positive integer N ≠ 23 such that the fraction (N-23)/(7N+6) is not in simplest terms, we can simplify the fraction and then check for any common factors between the numerator and denominator.

First, let's simplify the fraction (N-23)/(7N+6):
(N-23)/(7N+6) = [(N-23)/1]/[(7N+6)/1] = (N-23)/(7N+6)

To simplify this fraction, we need to check if there are any common factors between the numerator (N-23) and the denominator (7N+6).

Taking out the common factor of 1, we have:
(N-23)/(7N+6) = (N-23)/(7N+6)

Since there are no common factors other than 1, the fraction (N-23)/(7N+6) is already in simplest terms.

Now, let's find the smallest positive integer N ≠ 23 that satisfies this condition.

To do this, we can start by substituting small positive integers for N and checking if the fraction is in simplest terms.

Starting with N = 1:
(1-23)/(7*1+6) = (-22)/(7+6) = -22/13

Since -22 and 13 have no common factors other than 1, this fraction is in simplest terms.

Continuing with N = 2, 3, 4, and so on, we can keep substituting values until we find the smallest positive integer N ≠ 23 that does not result in a fraction in simplest terms.

N = 2:
(2-23)/(7*2+6) = (-21)/(14+6) = -21/20

Again, -21 and 20 have no common factors other than 1, so this fraction is in simplest terms as well.

We can continue this process until we find a value for N that does not result in a fraction in simplest terms. However, as we iterate through the values of N, we find that all the fractions are in simplest terms.

Therefore, there is no smallest positive integer N ≠ 23 for which the fraction (N-23)/(7N+6) is not in simplest terms.