2023 AMC 12B Problems/Problem 24
Contents
Problem
Suppose that , , and are positive integers satisfying all of the following relations.
What is ?
Solution
Denote by the number of prime factor in number .
We index Equations given in this problem from (1) to (7).
First, we compute for .
Equation (5) implies . Equation (2) implies . Equation (6) implies . Equation (1) implies .
Therefore, all above jointly imply , , and or .
Second, we compute for .
Equation (2) implies . Equation (3) implies . Equation (4) implies . Equation (1) implies .
Therefore, all above jointly imply , , and or .
Third, we compute for .
Equation (5) implies .
Equation (2) implies .
Thus, .
From Equations (5)-(7), we have either and , or and .
Equation (1) implies . Thus, for , , , there must be two 2s and one 0.
Therefore,
~Steven Chen (Professor Chen Education Palace, www.professorchenedu.com)
Video Solution
~Steven Chen (Professor Chen Education Palace, www.professorchenedu.com)
See Also
2023 AMC 12B (Problems • Answer Key • Resources) | |
Preceded by Problem 23 |
Followed by Problem 25 |
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 |
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