SOLUTION: A census taker stopped at a Hotel Sleep-Inn. The clerk at the desk was a mathematics student with a sense of humor. When asked how many guests were in the hotel, she replied, " The

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Question 871318: A census taker stopped at a Hotel Sleep-Inn. The clerk at the desk was a mathematics student with a sense of humor. When asked how many guests were in the hotel, she replied, " The number is the smallest positive integer that has the following properties:
When divided by two, the result is a perfect square:
When divided by three, the result is a perfect cube."
Can you figure out how many guests were in the hotel?
Answer by Edwin McCravy(20062)   (Show Source): You can put this solution on YOUR website!
To be divisible by both 2 and 3, it must be a number of the form



(since it needs no other prime factors to be as small as possible.)

When divided by two, the result is a perfect square:
So  is a perfect square.

Therefore m-1 and n are both even

Therefore m is odd and n is even.

When divided by three, the result is a perfect cube.
So  is a perfect cube.

Therefore m and n-1 are both multiples of 3.

Therefore m is a multiple of 3 and m is 1 more than a multiple of 3.

Putting everything together:

m is an odd multiple of 3.  

The smallest such positive integer is 3, so m=3

n is even and 1 more than a multiple of 3.  

The smallest such positive integer is 4, so n=4.

  

So there are 648 guests.

Checking:

648÷2 = 324 = 18²
648÷3 = 216 = 6³

Edwin
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