<s>[INST] Find all the numbers $n$ such that $\frac{4n-5}{60-12n}$ can't be reduced. 
Find all the numbers $n$ such that $\frac{4n-5}{60-12n}$ can't be reduced.

Attempt:
$$\gcd(4n-5,60-12n)=(4n-5,-8n+55)=(4n-5,-4n+50)=(4n-5,45)$$
$$n=1: (4-5,45)=1\quad \checkmark\\
n=2: (3,45)=3\quad \times\\
n=3: (7,45)=1\quad \checkmark\\
n=4: (11,45)=1\quad \checkmark\\
n=5: (15,45)=15\quad \times\\
n=6: (19,45)=1\quad \checkmark\\
n=7: (23,45)=1\quad \checkmark\\
n=8: (27,45)=9\quad \times\\
\vdots$$
So the answer is that it can't be reduced for $n=1,3,4,6,7,..$ i.e
$$\bigg\{n\bigg|n\notin \begin{cases}a_1=2\\a_n=a_{n-1}+3\end{cases}\bigg\}$$
I want to verify that my solution is correct
 [/INST] If $d$ is the common divisor of $4n-5$ and $60-12n$, then it should also divide $3(4n-5)+(60-12n) =45$. Thus when numerator and denominator are not divisible by 3 and 5, the fraction is irreducible. Hence $n \not \equiv 2 \pmod 3$ and $n \not \equiv 0 \pmod 5$.
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