Difference between revisions of "2016 AIME II Problems/Problem 8"
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Find the number of sets <math>{a,b,c}</math> of three distinct positive integers with the property that the product of <math>a,b,</math> and <math>c</math> is equal to the product of <math>11,21,31,41,51,61</math>. | Find the number of sets <math>{a,b,c}</math> of three distinct positive integers with the property that the product of <math>a,b,</math> and <math>c</math> is equal to the product of <math>11,21,31,41,51,61</math>. | ||
Revision as of 17:18, 22 March 2018
Problem
Find the number of sets 
of three distinct positive integers with the property that the product of 
and 
is equal to the product of 
.
Solution
Note that the prime factorization of the product is 
. Ignoring overcounting, by stars and bars there are 
ways to choose how to distribute the factors of 
, and ways to distribute the factors of the other primes, so we have 
ways. However, some sets have 
numbers that are the same, namely the ones in the form 
and 
, which are each counted times, and each other set is counted times, so the desired answer is 
.
Solution by Shaddoll
See also
| 2016 AIME II (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 | ||
| All AIME Problems and Solutions | ||
