A large integer is divisible by all the integers between 1 and 30 inclusive, except for two consecutive integers. Determine those two consecutive integers.
The two numbers must be 16 and 17.

First, note that if a number $n$ is not a power of a prime, then it has two smaller factors $a$ and $b$ that are relatively prime and multiply to $n$. Thus, $n$ is the least common multiple of $a$ and $b$. Then, any multiple of both $a$ and $b$ must also be a multiple of $n$, so any multiple of all the numbers less than $n$ would have to be a multiple of $n$ (since two of the numbers less than $n$ are $a$ and $b$ ). Thus, for there to be some number that is a multiple of all the numbers less than $n$ but not $n$ itself, $n$ must be a prime power. Therefore, both consecutive numbers are prime powers.

Note also that if $n$ is at most 15 , then $2 n \leq 30$. Thus, if we chose one of the consecutive numbers to be $n \leq 15$, the big number would have to be a multiple of $2 n$, and therefore also of $n$. Thus, this cannot happen, so both numbers must be more than 15. The only pair of consecutive numbers from 16 to 30 that are both prime powers is 16 and 17 , so this must be the answer.