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

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.)

So  is a perfect square.

Therefore m-1 and n are both even

Therefore m is odd and n is even.

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