If an integer $k$ is divisible by $2,5$ and $13$, what is the next largest number that is divisible by all the $3$ given numbers?

Find the required number based on given information

If a given number is divisible by a set of numbers, then it is either the LCM or a multiple of the LCM of the set of numbers.

According to given condition $k$ is divisible by $2,5,13$

Hence, $k$ is either the LCM or a multiple of the LCM of the numbers $2,5,13$

$2,5$ and $13$ are co prime numbers.

So the LCM of these numbers is given by the product of these numbers

Therefore, LCM$=2\times 5\times 13=130$

We need to find the next largest number that is divisible by $2,5$ and $13$ which is the next multiple of $130$ after $k$

Hence, the required number is equal to $k+130$.