Solution: Nth Magical Number

Statement

Given three integers n, a, and b, return the nth magical number.

A magical number is defined as a positive integer that is divisible by either a or b.

As the result may be very large, return it modulo $10^9+7$.

Constraints:

• $1 \leq$ n $\leq 10^9$

• $2 \leq$ a, b $\leq 4 \times 10^4$

Solution

To find the nth number divisible by either a or b, we observe that magical numbers form a sorted sequence of all multiples of a and b. Instead of generating this sequence explicitly, we can efficiently count how many magical numbers are $\leq x$ using the inclusion-exclusion principle:

• Count of numbers divisible by a: $\lfloor x / a \rfloor$

• Count of numbers divisible by b: $\lfloor x / b \rfloor$

• Subtract numbers divisible by both (i.e., $\lfloor x / lcm(a, b) \rfloor$) to avoid double counting.

Where the least common multiple (LCM) of two numbers, $a$ and $b$, is the smallest number divisible by both, and is given by: