2002 AMC 10P Problems/Problem 8
Problem
How many ordered triples of positive integers 
satisfy 
Solution 1
The given expression, 
, is equivalent to 
. Next, notice how 
must be a power of 
less than 
and the exponent of its prime factorization must be a factor of 
otherwise, 
won't be an integer. We can use a bit of case work to solve this problem.
Case 1: 
Clearly, our only four solutions are 
and 
along with 
and 
Case 2: 
Clearly, our only two solutions are 
and 
Case 3: 
Clearly, our only two solutions are 
and 
Case 4: 
Clearly, our only solution is 
Adding up all our cases gives 
Thus, our answer is \boxed{\textbf{(E) }9}$.
See also
| 2002 AMC 10P (Problems • Answer Key • Resources) | ||
| Preceded by Problem 7 |
Followed by Problem 9 | |
| 1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25 | ||
| All AMC 10 Problems and Solutions | ||
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