The problem is as follows:
On the occasion of Bob's grand opening of his new restaurant he noticed that he only has $10$ VIP tables. These will be used to accomodate $48$ men and $36$ women. He wants to place them in such a way that, using the largest possible number of tables, each table has the same number of men and the same number of women. Find the difference between the number of men and women at each table.
The alternatives given in my book are as follows:
$\begin{array}{ll} 1.&\textrm{2}\\ 2.&\textrm{1}\\ 3.&\textrm{3}\\ 4.&\textrm{10}\\ \end{array}$
The official solution from my book is to use the quantity of divisors for 45 and for 36.
It alludes that the number of people to be seated in each table is the same but it doesn't necessarily imply that the number of men is the same as the number of women per table.
The notation $QD$ indicates the quantity or the number of divisors from that number.
It states the following:
$\operatorname{factor}(48)=2^4\cdot 3 $
$\operatorname{QD}(45)=(4+1)(1+1)=10$
$\operatorname{factor}(36)=3^2\cdot 2^2$
$\operatorname{QD}(36)=(2+1)(2+1)=9$
Then it goes to say that:
$48=6(8)$
$46=6(6)$
From this it concludes that there are $8$ men to be seated in the VIP table and $6$ women would be seated in the VIP table.
And since it is being asked the difference, it declares that:
$8-6=2$ hence the answer is the first option.
But I'm stuck. I don't know the logic for this process. Why the author did that?.
There are skipped steps here to my understanding. What does this has to do with the quantity of divisors for $48$ and $36$?
Can someone help me here?. Does it exist a better way to solve this problem perhaps using a similar approach or an easier one?.
Does this problem require the use of least common multiple or greatest common divisor?. Which one and why?
I'm lost here, it would really help me if someone could include a step by step explanation on how to solve this without much fuss, but more importantly without skipping any steps because I'm lost.
Let $T$ tables be used. Since there are equal numbers of men on each table, $T$ must divide $48$. Similarly $T$ must divide $36$.
Therefore $T$ divides the gcd $(48,36)$ which is $12$ . Hence, since we only have $10$ tables, there are $6$ tables.
You have clearly understood the rest of the argument.