If an integer 'k' is divisible by 2, 5, and 13, what is the next number after…

Concept: If a number must be divisible by several given integers at once, then consecutive numbers satisfying that condition are spaced apart by exactly the Least Common Multiple (LCM) of those integers — the very next such number after any qualifying value is obtained by adding their LCM to it.

Application:

1. k is divisible by 2, 5, and 13, so k is already a common multiple of all three.

2. 2, 5, and 13 are pairwise coprime (all prime), so their LCM is their direct product: LCM(2, 5, 13) = 2 × 5 × 13 = 130.

3. The next integer after k that is again divisible by 2, 5, and 13 must be k plus this LCM, since any smaller positive addition cannot be guaranteed divisible by all three for every possible k.

Cross-check: Taking k = 260 (itself divisible by 2, 5 and 13), the next common multiple is 390 = 260 + 130, confirming the step size of 130 — notice this is also distinct from 2k = 520, a different, larger value.

Result: The next number divisible by 2, 5 and 13 after k is k + 130.