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

To find the smallest positive integer N ≠ 23 such that the fraction \(\frac{N-23}{7N+6}\) is not in simplest terms, we need to simplify this fraction for different values of N until we find one that cannot be simplified further.

Let's go through the process step by step:

1. Start by factoring the numerator \(N-23\) and the denominator \(7N+6\) if possible:
\(N-23\) cannot be factored further.
\(7N+6\) cannot be factored further.

2. Since we cannot simplify the numerator or the denominator, we can proceed to check if both share any common factors.

3. To check if both share any common factors, we calculate their greatest common divisor (GCD).
GCD(7N+6, N-23) = GCD(7, N-23) = 1

4. If the GCD of the numerator and denominator is 1, it means they are coprime (no common factors other than 1) and the fraction is already in its simplest form. In this case, we need to find the smallest N ≠ 23 that does not yield a GCD of 1.

5. Start testing different values for N greater than 23 and calculate the GCD for each value until we find one that does not yield a GCD of 1.

a. Let's test N = 24:
GCD(7N+6, N-23) = GCD(7*24+6, 24-23) = GCD(174, 1) = 1

b. Let's test N = 25:
GCD(7N+6, N-23) = GCD(7*25+6, 25-23) = GCD(181, 2) = 1

c. Let's test N = 26:
GCD(7N+6, N-23) = GCD(7*26+6, 26-23) = GCD(188, 3) = 1

d. Let's test N = 27:
GCD(7N+6, N-23) = GCD(7*27+6, 27-23) = GCD(195, 4) = 1

e. Let's test N = 28:
GCD(7N+6, N-23) = GCD(7*28+6, 28-23) = GCD(202, 5) = 1

f. Let's test N = 29:
GCD(7N+6, N-23) = GCD(7*29+6, 29-23) = GCD(209, 6) = 1

g. Let's test N = 30:
GCD(7N+6, N-23) = GCD(7*30+6, 30-23) = GCD(216, 7) = 7

6. We can see that for N = 30, the GCD is not 1, which means the fraction \(\frac{N-23}{7N+6}\) is not in its simplest form.

Therefore, the smallest positive integer N ≠ 23 that results in a fraction not in simplest terms is N = 30.