Suppose all are right..

then the number would be

Now those who cannot be wrong are- 

(1,2),(2,3),(3,4),(4,5),(5,6),(6,7),(7,8),.......(15,16),   (17,18),(18,19),(19,20)....(24,25)

So it has to be (16,17)

The number is therefore

=787386600..

the LCM of the first 25 positive integers is

2^4 * 3^2 * 5^2 * 7 * 11 * 13 * 17 * 19 * 23...

first of all we have to remove two consecutive positive integers.

we will start from the terms from the back.

24*25. but then we already have 5 and 15 and than will mean that we cant take out 25.

since the number is divisible by 5 and 15, it is also divisible by 25.

similalry if 24*23 isnt possible too since we have 8 and 3.

since the number is divisible by 8 and 3 it has to be divisible by 24...

so continue like this. it will be easier to look for primes and the numbers next to them.

we reach 16 and 17.

It is the only number <25 that has four factors of two.
Thus if we say that it isnt divisible by 16, we are saying that it doesnt have four factors of 2. But no other numbers do either, and thus it is legal.