False Divisors
Logic · proposed by @Th4nh
A teacher writes a positive integer $N$ on the board. Twelve students, numbered 1 through 12, each make a statement about the number:
Student 1: "$N$ is divisible by 1."
Student 2: "$N$ is divisible by 2."
Student 3: "$N$ is divisible by 3."
...and so on, up to:
Student 12: "$N$ is divisible by 12."
The teacher reveals: "Exactly two of the statements made by the students are false, and those two students spoke consecutively."What is the smallest possible value of $N$?