Permutations , Probablity and Combinations

Question

A group of 25 friends were discussing a large positive integer. "It can be divided by 1," said the first friend. "It can be divided by 2," said the second friend. "And by 3," said the third friend. "And by 4," added the fourth friend. This continued until everyone had made such a comment. If exactly two friends were incorrect, and those two friends said consecutive numbers, what was the least possible integer they were discussing?

Answer 1

Note that if the friend who said 3 is wrong, then any friends who say any multiple of 3 (such as 6 or 9) will also be wrong. This limits the consecutive integers to be greater than 12 (so their multiples are not in 25).

If the friend who said 6 is wrong, then there the friend who said 2 is wrong or the friend who said 3 is wrong, and 2 or 3 is not consecutive to 6. This limits the integers to be powers of a single integer or primes.

We can easily find the two consecutive integers following the above conditions by checking. They are 16 and 17.

Thus, the integer is LCM(1,2,...,15,18,19,...,25)=787386600.

Answer 2

Any number is divisible by $1$ therefore, divisible by $1$ is a true statement. So that number must be divisible by $2$ and so the number is $2$.