2003 AMC 12B Problems/Problem 18
Contents
Problem
Let and be positive integers such that The minimum possible value of has a prime factorization What is
Solution 1
Substitute into . We then have . Divide both sides by , and it follows that:
Note that because and are prime, the minimum value of must involve factors of and only. Thus, we try to look for the lowest power of such that , so that we can take to the fifth root. Similarly, we want to look for the lowest power of such that . Again, this allows us to take the fifth root of . Obviously, we want to add to and subtract from because and are multiplied by and divided by , respectively. With these conditions satisfied, we can simply multiply and and substitute this quantity into to attain our answer.
We can simply look for suitable values for and . We find that the lowest , in this case, would be because . Moreover, the lowest should be because . Hence, we can substitute the quantity into . Doing so gets us:
Taking the fifth root of both sides, we are left with .
Solution 2
A simpler way to tackle this problem without all that modding is to keep the equation as:
As stated above, and must be the factors 7 and 11 in order to keep at a minimum. Moving all the non-y terms to the left hand side of the equation, we end up with:
The above equation means that must also contain only the factors 7 and 11 (again, in order to keep at a minimum), so we end up with:
( and are arbitrary variables placed in order to show that could have more than just one 7 or one 11 as factors)
Since 7 and 11 are prime, we know that and . The smallest positive combinations that would work are and . Therefore, . is correct.
See Also
2003 AMC 12B (Problems • Answer Key • Resources) | |
Preceded by Problem 17 |
Followed by Problem 19 |
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 12 Problems and Solutions |
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.