2008 iTest Problems/Problem 29
Problem
Find the number of ordered triplets of positive integers such that (the product of , and is ).
Solution
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.
See Also
2008 iTest (Problems) | ||
Preceded by: Problem 28 |
Followed by: Problem 30 | |
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