What are the Two Integers?

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

1, 2, 3, \dots, 10000,

(1)

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.

Assume the question is valid, i.e., there is a unique answer to this problem.
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.
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 $2^{13}=8192$ .
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.