2008 iTest Problems/Problem 29
Contents
Problem
Find the number of ordered triplets of positive integers such that
(the product of
, and
is
).
Solution 1
The number can be factored into
. Use casework to organize the counting.
- If two numbers are
, then the third one must be
, and there are
ways to write the ordered pairs.
- If one number is a
, then there are
possible pairs of numbers for the other two. Since the numbers are all different, there are
ways to write the ordered pairs.
- If none of the numbers are
, then since there are only four prime numbers being multiplied, one of the numbers must have two prime numbers being multiplied together. Thus, the two sets of numbers are
and
, and there are
ways in this case.
Altogether, there are ordered pairs that satisfy the criteria.
Solution 2
can be prime factorized into
. We can think of each ordered pair
as a way to assign three 2s and one 251 to three distinct letters. You may now recognized this as a "assign non-distinct objects to distinct piles" problem. In problems like this, we should use stars and bars. There are
ways to assign three 2s to three distinct letters, and there are
ways to assign one 251 to three distinct letters. Multiplying, we get
.
See Also
2008 iTest (Problems) | ||
Preceded by: Problem 28 |
Followed by: Problem 30 | |
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