Math Problem Statement

Let B, C, and O be distinct positive integers such that the product B * C * O = 2014. What is the largest possible value of the sum B + C + O?

Solution

To solve this problem, we need to maximize the sum B+C+OB + C + O under the condition that BB, CC, and OO are distinct positive integers and their product equals 2014.

Step 1: Factorize 2014

We start by finding the prime factorization of 2014: 2014=2×19×532014 = 2 \times 19 \times 53 So, the product B×C×O=2014B \times C \times O = 2014 implies that BB, CC, and OO must be chosen as distinct positive integers whose product equals 2×19×532 \times 19 \times 53.

Step 2: Choosing BB, CC, and OO to Maximize B+C+OB + C + O

To maximize the sum B+C+OB + C + O, 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 BB, CC, and OO are:

  • B=1B = 1, C=2×19=38C = 2 \times 19 = 38, O=53O = 53
  • This setup satisfies B×C×O=2014B \times C \times O = 2014 and gives distinct values for BB, CC, and OO.

Calculating the sum: B+C+O=1+38+53=92B + C + O = 1 + 38 + 53 = 92

Conclusion

The largest possible value of B+C+OB + C + O is: 92\boxed{92}

Would you like further details on any part of this solution?

Related Questions

  1. How would the solution change if BB, CC, and OO could be non-distinct integers?
  2. How can we approach similar problems with a different target product, like 1000 instead of 2014?
  3. Why is it beneficial to factorize the target number to solve such product-sum problems?
  4. What are some strategies to maximize sums in integer factorization problems?
  5. How can the solution be adapted if BB, CC, and OO 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