Math Problem Statement
- Consider the system of equations
20[x/2]+[y/2]=103 ,21[x/2]+[y/2]=109
where and y are positive integers. Compute the least possible value of x + y.
Solution
To solve the given system of equations, we first rewrite them:
We subtract the first equation from the second:
Simplifying, we get:
With , it implies:
Next, we substitute back into one of the original equations to find . Using the first equation:
There appears to be an error here. Let's carefully substitute again:
Using :
Clearly, there's an inconsistency because cannot be negative. Let's retry the substitution carefully:
Using the correct :
So,
Thus, the minimum :
Thus, the least possible value of is:
Do you want further details or have any questions? Here are some questions you might ask next:
- Can you explain why was incorrect?
- How would we solve for and if there were different equations?
- What is the definition and properties of the floor function?
- Can you show other examples where we solve for integer solutions in similar equations?
- What if and were not integers, how would the solution method change?
Tip: When working with floor functions, always consider the ranges carefully to avoid inconsistencies.
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Math Problem Analysis
Mathematical Concepts
Floor function
System of equations
Formulas
-
Theorems
-
Suitable Grade Level
Grades 9-12
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