How many five-digit numbers can be formed using digits $1,2,3,4,5,6$ which are divisible by $3$ and $5$? How many five-digit numbers can be formed using digits $1,2,3,4,5,6$ which are divisible by $3$ and $5$, without any of the digits repeating?
A number is divisible by $5$ if and only if last digit is $0$ or $5$ and; divisible by $3$ if and only if the sum of the digits add to a multiple of $3$ ($3,6,9,12,15,18,21$).
Divisible by $5$ means last digit is $5$. That's the only option.
Divisible by $3$ means that the sum of the digits add up to a multiple of $3$.
What can we do next? It's difficult for me to find the 5-tuples just by looking at the given digits? What can I do to systematize them and not miss some?

hint
The mostright digit should be $ 5$.
now, we have to choose $ 4 $ digits from $5 $. there are five possibilities
$$\{1,2,3,4\}\;;\;\{1,3,4,6\}\;;\;\{1,2,4,6\}\;;\;\{2,3,4,6\}\;;\;\{1,2,3,6\}$$
take those for which the (sum +5) is divisible by $ 3$.
we find
$$\{1,2,4,6,5\}\;;\;\{1,2,3,4,5\}$$
The final answer is $2\times 4!=48$.
