
When tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.

[ASSESS]

[ADVANCE]

[VERIFY]

[SIMPLIFY]

[SYNTHESIZE]

[PIVOT]

[OUTPUT]

You should strictly follow the format below:

[ACTION NAME]

# Your action step 1

# Your action step 2

# Your action step 3

...

Next action: [NEXT ACTION NAME]


system
A fraction is called irreducible when its numerator and denominator have no common factors, that is, when the greatest common divisor (GCD) of the two numbers is 1. For example, the fraction \(\frac{3}{7}\) is irreducible, but the fraction \(\frac{10}{14}\) is not, since 2 is a common factor of 10 and 14. For which values of \(n\) is the fraction \(\frac{5n+6}{6n+5}\) irreducible? Let's study this problem in parts:

a) Let \(d = \text{GCD}(5n+6, 6n+5)\) be the greatest common divisor of \(5n+6\) and \(6n+5\). Verify that \(d\) is a divisor of \(n-1\).

b) Knowing that \(d\) is a divisor of \(n-1\), conclude that \(d\) is also a divisor of 11.

c) Verify that if 11 divides \(5n+6\), then 11 divides \(6n+5\).

d) For how many positive integers \(n\), less than 50, is the fraction \(\frac{5n+6}{6n+5}\) irreducible?

Present the answer in LaTex format: \boxed{Your answer}
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