For a positive integer n > 2, consider the n-1 fractions
2/1; 3/2; ...; n/(n-1).
The product of these fractions equals n, but if you reciprocate (i.e. turn upside down) some of the fractions, the
product will change.
How can you make the product equal 1?
Find all values of n for which this is possible.
Paul gave the solution for n=4: 1/2*3/2*4/3=1
notice the primes. 2 and 2 cancel the 4 and the 3's cancel each other.
If you go to n=5 you introduce a new prime that does not cancel, you'd need to increase n to 10 to get another 5 but this just introduces more primes (7), each of which will have the same problem. Since between any prime p and 2p there is another prime (thanks Erdös), no higher n will work.
|
|
Posted by Jer
on 2015-12-02 07:18:54 |
Different solution
