Math Problem Statement
Solution
To solve this problem, we need to maximize the sum under the condition that , , and are distinct positive integers and their product equals 2014.
Step 1: Factorize 2014
We start by finding the prime factorization of 2014: So, the product implies that , , and must be chosen as distinct positive integers whose product equals .
Step 2: Choosing , , and to Maximize
To maximize the sum , we want to distribute the factors in a way that gives the largest possible values while keeping them distinct. Given the prime factors, the possible values of , , and are:
- , ,
- This setup satisfies and gives distinct values for , , and .
Calculating the sum:
Conclusion
The largest possible value of is:
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Related Questions
- How would the solution change if , , and could be non-distinct integers?
- How can we approach similar problems with a different target product, like 1000 instead of 2014?
- Why is it beneficial to factorize the target number to solve such product-sum problems?
- What are some strategies to maximize sums in integer factorization problems?
- How can the solution be adapted if , , and were constrained to be consecutive numbers?
Tip: Always start by factorizing the given number in problems involving products of integers. This often reveals useful insights and possible configurations for the solution.
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Math Problem Analysis
Mathematical Concepts
Number Theory
Factorization
Optimization of Sums
Formulas
Product Formula: B * C * O = 2014
Theorems
Prime Factorization
Suitable Grade Level
Grades 10-12
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