By Thomas Keilen
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Additional resources for Algebra I [Lecture notes]
And now settle the guests according to squares," the mathematician-philatelist said. " The director did not understand. "By squares. , from the second room of the first hotel ; in number 3 put the guest from ( 2,2), the second room of the second hotel , and in number 4 - the guest from ( 2, 1 ), the first room of the second hotel . We will thus have settled the guests from the upper left square of side 2. After this, put the guest from ( 1 ,3) in number 5, from ( 2,3) in number 6, from ( 3,3) in number 7, from ( 3,2) in number 8, from ( 3, 1 ) in number 9.
The fourth into numbers 7, 49, 343, . . " the director asked. "No. After all , if you take two prime numbers, none of their positive integer powers can equal one another. " The director agreed with me and immediately found an i mprovement on the method I had proposed, in which only the primes 2 and 3 were needed. Namely, he proposed to put the guest from the mth room of the nth hotel into room number 2m 3 n • This works because if m =1= p or n =1= q, 2m 3 n =1= 2P 3Q • SO no room would have two occupants.
I've had enough ! " the director shouted. "First I put up one guest in an already full hotel, then another 999,999, then even an infinite set of guests ; and now they want me to find room in it for an additional infinite set of infinite sets of guests. " But an order was an order, and they had five days to get ready for the arrival of the new guests. Nobody worked in the hotel during these five days everybody was pondering how to solve the problem. A contest was announced - the prize would be a tour of one of the galaxies.