Find the third GCD by knowing two other greatest common divisors

Question

The greatest common divisor of a natural number $n$ and $90$ equals 18. The GCD of $n$ and $120$ equals 12. How can I find the GCD of $n$ and $900$?

Answer 1

\begin{align} \gcd(n,90)=18 &\implies n=18p \\ &\implies \gcd(p,5)=1 \end{align}

\begin{align} \gcd(n,120)=12 &\implies n=12q \\ &\implies \gcd(q,10)=1 \\ &\implies \gcd(q,2)=\gcd(q,5)=1 \end{align}

\gcd(n,900) = \gcd(18p, 900) = 18 \gcd(p, 50) = 18 \gcd(p,2) \in {18, 36}$

$\gcd(n,900) = \gcd(12q, 900) = 12 \gcd(q, 75) = 12 \gcd(q,3) \in \{12, 36\}$

So $\gcd(n,900) = 36$