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.
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Assume the question is valid, i.e., there is a unique answer to this problem.
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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.
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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 \(2^{13}=8192\).
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The other number must be either
8193
or8191
. Since8193
is divisible to3
but not9
(8 + 1 + 9 + 3 = 21
is a multiple of3
but not9
), it does not have the form mentioned in step 2. So the other number is8191
.