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Chapter 13: Problem 2

Devise an algorithm to solve this problem in polynomial time. The input is a composite integer \(n\), not a power, and a proper factor \(a\) of \(n\). The output consists of two relatively prime integers \(c, d\) satisfying \(n=c d\) and \(1

Short Answer

Expert verified
The algorithm is as follows: 1. Identify the given input variables, 2. Determine factor pairs of \(n\), 3. Derive relatively prime pairs, 4. Verify and choose the solution pair. The result will be the pair, \(c\) and \(d\), which are relatively prime and satisfy the given conditions

Step by step solution

01

Identify given input variables

The given input variables are a composite integer \(n\), and a proper factor \(a\) of \(n\). The composite number \(n\) is not a power.
02

Determine factor pairs of n

Start by finding the factor pairs of \(n\). Pairing the factors can ensure the formulation of \(c\) and \(d\). Factor pairs are pairs of numbers that, when multiplied together, give the original number \(n\). You need to divide \(n\) by increasing values starting from 1 to find these pairs.
03

Derive relatively prime pairs

Relatively prime or coprime numbers are two numbers which have no common divisor other than 1. From the factor pairs found in Step 2, identify pairs that are relatively prime. You want to find a pair such that they satisfy the equation \(n = c \cdot d\) and \(1 < c < n\).
04

Verify and choose the solution pair

Now that you have a list of relatively prime factor pairs, you can go through each pair and check if the pair satisfies all given conditions. Once found, \(c\) and \(d\) are your algorithm's output.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Composite Number
A composite number is an integer greater than one that is not prime. Unlike prime numbers, which have only two divisors鈥�1 and itself, composite numbers have more than two divisors.
For example, the number 15 is composite because it can be divided by 1, 3, 5, and 15.
  • Composite numbers always have at least one divisor besides 1 and itself.
  • They can be expressed as the product of smaller integers, which are its factors.
  • Examples of composite numbers are 4, 6, 8, 9, 10, etc.
Understanding composite numbers is essential when breaking down numbers into their factors, which is a key element in solving many algorithmic problems.
Factor Pairs
Factor pairs are two numbers that, when multiplied together, give back the original number. They are foundational in finding the way numbers can be split into parts or products.
For example, if we consider the composite number 12, its factor pairs are:
  • (1, 12)
  • (2, 6)
  • (3, 4)
When breaking down a composite number into factor pairs, each unique pair reflects a possible way the number can be decomposed. Finding factor pairs is crucial in algorithmic problem solving, especially when needing to find values that work together under certain constraints.
Relatively Prime
Relatively prime numbers, also known as coprime numbers, are two numbers that have no common divisors other than 1.
For instance, 8 and 15 are relatively prime because their only common divisor is 1.
  • Relatively prime pairs play an important role when the goal is to find numbers that multiply to a given number with certain conditions.
  • Checking for common divisors can be done using the Euclidean algorithm, which efficiently determines the greatest common divisor (GCD) of two numbers.
Understanding how to identify relatively prime pairs helps in many areas, such as simplifying fractions, or in this case, splitting a composite number into desired products.
Algorithmic Problem Solving
Algorithmic problem solving is a method of approaching problems in a structured manner to find efficient solutions. It involves creating step-by-step instructions that can be executed by a computer to achieve a desired outcome.
In the context of breaking down composite numbers, algorithmic problem solving is applied to:
  • Identify all factors of a composite number.
  • Check combinations of factors to find pairs that fulfill given conditions, such as being relatively prime.
  • Ensure the problem is solved within a specific time complexity, like polynomial time.
This methodical approach is critical in determining solutions that are both correct and efficient, particularly in areas related to cryptography, numerical analysis, and optimization tasks.

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Factor Kraitchik's number \(193541963777 .\)
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