1995 AHSME Problems/Problem 29
Contents
Problem
For how many three-element sets of distinct positive integers is it true that ?
Solution 1
. We wish to figure out the number of ways to distribute these prime factors amongst 3 different integers, without over counting triples which are simply permutations of one another.
We can account for permutations by assuming WLOG that contains the prime factor 2. Thus, there are ways to position the other 4 prime numbers. Note that, with the exception of when all of the prime factors belong to , we have over counted each case twice, as for when we put certain prime factors into and the rest into , we count the exact same case when we put those prime factors which were in into .
Thus, our total number of cases is
Solution 2
The prime factorization of is Therefore, we have the equation where must be distinct positive integers and order does not matter. There are ways to assign each prime number on the right-hand side to one of the variables which gives solutions for However, three of these solutions have two s and one which contradicts the fact that must be distinct. Because each prime factor appears only once, all other solutions have distinct. Correcting for this, we get ordered triples where are all distinct.
Finally, since order does not matter, we must divide by the number of ways to order This gives the final answer,
See also
1995 AHSME (Problems • Answer Key • Resources) | ||
Preceded by Problem 28 |
Followed by Problem 30 | |
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