2001 AMC 12 Problems/Problem 21
Contents
Problem
Four positive integers , , , and have a product of and satisfy:
What is ?
Solution 1
Using Simon's Favorite Factoring Trick, we can rewrite the three equations as follows:
Let . We get:
Clearly divides . On the other hand, can not divide , as it then would divide . Similarly, can not divide . Hence divides both and . This leaves us with only two cases: and .
The first case solves to , which gives us , but then . (We do not need to multiply, it is enough to note e.g. that the left hand side is not divisible by .) Also, a - d equals in this case, which is way too large to fit the answer choices.
The second case solves to , which gives us a valid quadruple , and we have .
Solution 2
As above, we can write the equations as follows:
Looking at the first two equations, we know that but not is a multiple of 5, and looking at the last two equations, we know that but not must be a multiple of 5 (since if or was a multiple of 5, then would also be a multiple of 5).
Thus, , and . The only answer choice where this is true is .
See Also
2001 AMC 12 (Problems • Answer Key • Resources) | |
Preceded by Problem 20 |
Followed by Problem 22 |
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