I met a friend majored in math on a bus home today. He held a piece of paper with a question (probably an interview question since he is trying to find a job recently). He asked the question to me and I found it to be an interesting one.

A very large positive integer is divisible by all but two of the integers

and the two excepted numbers are consecutive integers. What are the two integers?

I did not get the answer before my friend get off the bus. However, as soon as arriving home I get the key to the questions. The outline of my thoughts leading to the answer is as follows.

1. Assume the question is valid, i.e., there is a unique answer to this problem.

2. Both of the two numbers (indivisible to the large integer) have prime factorization of the form: \(a^b\), where \(b\) is the largest possible value.

3. Since one of two consecutive integers is even, one of the two numbers (indivisible to the large integer) has the form \(2^b\) and thus is \(2^{13}=8192\).

4. The other number must be either 8193 or 8191. Since 8193 is divisible to 3 but not 9 (8 + 1 + 9 + 3 = 21 is a multiple of 3 but not 9), it does not have the form mentioned in step 2. So the other number is 8191.