Question about gcd (greatest-common-divisor)

Question

I have this question about gcd and I have doubts:

"The gcd between two numbers is 48 and the largest number is 384. What is the other number?"

I thought that the missing number must be a multiple of 48 as well, but I can't find a way to find that number.

Thanks.

Answer 1

The other number is of the form $48k$ with $\gcd(k,\tfrac{384}{48})=1$. Given that $384$ is the largest of the two, we have $k\leq\tfrac{384}{48}=8$. This leaves $k=1,3,5,7$, assuming $k$ is positive, yielding $4$ solutions.

Answer 2

I don't recommend this method in general, but maybe it gives you some intuition:

You can decompose $384=2^7·3$. If you divide by $48=2^4·3$ you are left with $2^3$. This means that the other number cannot have more $2$s than the ones from $48$ in its prime decomposition, or otherwise the $\gcd$ with $384$ would be greater. Therefore you can multiply $48$ by $1$; not $2$; $3$; $5$; not by $6$ since it would be multiplying by $2$ too; $7$; not $8$, and we have to stop there because $48·9>384$.

So there are four solutions: $48·1=48$; $48·3=144$; $48·5=240$; $48·7=336$.