I am struggling to solve this problem and would appreciate any help:
When is $\frac{12x+5}{12y+2}$ NOT in lowest terms? ($x$,$y$ are nonnegative integers)
I have found that it is not in lowest terms for $x=6$ and $y=9$ because numerator and denominator are divisible by $11$, but I'm stuck here.
EDIT: Apparently "lowest terms" isn't in common usage in maths, so I will have to explain what it means. A fraction $p/q$ with $p,q\in \mathbb{Z}$ and $q\ne 0$ is in lowest terms when $\gcd(p,q)=1$. Otherwise, it is not in lowest terms.
For example, $\frac{3}{5}$ and $\frac{9}{2}$ are in lowest terms, but $\frac{15}{3}$ and $\frac{17}{34}$ are not.
Fractions in general are in lowest terms $6/\pi^2$ of the time.
One quarter of fractions can cancel 2, one ninth can cancel 3, and so on, so the proportion with no prime factor to cancel is $$\frac34\cdot\frac89\cdot\frac{24}{25}\cdot\frac{48}{49}\cdots=\frac6{\pi^2}$$
For these fractions, $2$ and $3$ will never cancel, so we lose the factors of $3/4$ and $8/9$ from the left-hand side.
These fractions are in lowest terms this proportion of the time: $$\frac43\frac98\frac6{\pi^2}=\frac9{\pi^2}\approx 0.91189$$