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Chapter 4: Problem 12

Let \(a, b \in \mathbf{Z}^{+}\)where \(a \geq b\). Prove that \(\operatorname{gcd}(a, b)=\) \(\operatorname{gcd}(a-b, b)\)

Short Answer

Expert verified
The statement \( \operatorname{gcd}(a, b)= \operatorname{gcd}(a-b, b)\) is proved to hold true for all positive integers \(a\) and \(b\) where \(a \geq b\). The proof is conducted using the Euclidean algorithm and proving by contradiction.

Step by step solution

01

Statement of Problem

Given positive integers \(a\) and \(b\) with \(a \geq b\), prove that \(\operatorname{gcd}(a, b)= \operatorname{gcd}(a-b, b)\).
02

Assume the Order Pair

Assume that \((a, b)\) is a pair of positive integers such that \(\operatorname{gcd}(a, b) \neq \operatorname{gcd}(a-b, b)\). Choose such a pair \((a, b)\) such that \(a + b\) has the smallest possible value.
03

Express GCD as a Linear Combination

For any pair of integers \(a\) and \(b\), the gcd can be expressed as a linear combination i.e., \(\operatorname{gcd}(a, b) = sa + tb\) for some integers \(s\) and \(t\).
04

Prove by Contradiction

Suppose, for contradiction, that \(\operatorname{gcd}(a, b)= d\), \(\operatorname{gcd}(a - b, b) = d'\), and \( d \neq d'\). Then, we can write \(d=sa+tb\) and \(d'=s'(a-b)+t'b\) for some integers \(s\), \(t\), \(s'\), and \(t'\).
05

Manipulate the Second Equation

Notice that we can rewrite the equation \(d'=s'(a-b)+t'b\) as \(d'=(s's+t')a-t'b\).
06

Apply the Euclidean Algorithm

Since d' divides \(a\) and \(b\), it should also divide \((s's+t')a-t'b\). Thus, \(d'\) must divide \(d\). Since \(d'=s'(a-b)+t'b\), any divisor of \(a-b\) and \(b\) must also divide \(d\). Therefore, \(d'\) is a divisor of \(a\), \(b\) and \(d\).
07

Contradiction

But according to our assumption \(d'\) is greater than \(d\), resulting in contradiction because a divisor cannot be greater than the number itself.
08

Counter the Initial Assumption

This contradiction is due to the false assumption that there exists a pair \((a, b)\) such that \( \operatorname{gcd}(a, b) \neq \operatorname{gcd}(a-b, b). Therefore, it is true for all positive integers \(a\) and \(b\) with \(a \geq b\) that \( \operatorname{gcd}(a, b)= \operatorname{gcd}(a-b, b)\).

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Key Concepts

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

Understanding the Greatest Common Divisor (GCD)
The Greatest Common Divisor (GCD) of two numbers is the largest number that divides both of them without leaving a remainder. It's a fundamental concept in number theory and a building block for more advanced math topics.
To find the GCD of two numbers, you can use the Euclidean algorithm, which involves repeated subtraction or division. This method recursively applies the principle that the GCD of two numbers is the same as the GCD of the smaller number and the remainder when the larger number is divided by the smaller number.
The GCD has practical applications too:
  • Simplifying fractions to their simplest form.
  • Solving problems involving divisibility.
  • Helping in algorithms for computing modular inverses in cryptography.
By understanding the properties of the GCD, one can solve various mathematical problems effectively.
Proof by Contradiction
Proof by contradiction is a clever and powerful method in mathematics to establish the truth of a statement. It involves assuming that the statement we want to prove is false, and then showing that this assumption leads to a logical contradiction. Once a contradiction is found, the original statement must be true.
Here's a simplified approach to how it works:
  • Assume that the statement is false.
  • Logically proceed with this assumption to reach an inconsistency or contradiction.
  • Since a contradiction is encountered, conclude that the original assumption must be wrong.
This method is particularly useful in proving results about the greatest common divisor because it allows us to eliminate any exceptions to a rule by showing they would lead to impossible situations.
Concept of Linear Combination
The concept of a linear combination is an essential tool in algebra and number theory. A linear combination of two integers involves expressions of the form \( sa + tb \) where \( s \) and \( t \) are integers.
This expression signifies that one number can be formed by adding multiples of two other numbers. It's particularly relevant to the GCD because:
  • Any GCD of two numbers can be expressed as their linear combination.
  • This expression helps in proving relationships between numbers, such as divisibility.
For instance, if \( \ ext{gcd}(a, b) = d \), then \( d \) can be expressed as a linear combination \( sa + tb = d \). This equation is foundational for proofs involving the Euclidean algorithm and supports concepts like Bezout's identity. By expressing the GCD this way, we can neatly solve problems involving number theory and algebra.

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Most popular questions from this chapter

Consider the permutations of \(1,2,3,4\). The permutation 1432, for instance, is said to have one ascent - namely, 14 (since \(1<4)\). This same permutation also has two descents namely, 43 (since \(4>3)\) and 32 (since \(3>2\) ). The permutation 1423 , on the other hand, has two ascents, at 14 and 23 - and the one descent 42 . a) How many permutations of \(1,2,3\) have \(k\) ascents, for \(k=0,1,2 ?\) b) How many permutations of \(1,2,3,4\) have \(k\) ascents, for \(k=0,1,2,3 ?\) d) Suppose a permutation of \(1,2,3, \ldots, m\) has \(k\) ascents, for \(0 \leq k \leq m-1\). How many descents does the permutation have? e) Consider the permutation \(p=12436587\). This permutation of \(1,2,3, \ldots, 8\) has four ascents. In how many of the nine locations (at the start, end, or between two numbers) in \(p\) can we place 9 so that the result is a permutation of \(1,2,3, \ldots, 8,9\) with (i) four ascents; (ii) five ascents? f) Let \(\pi_{m, k}\) denote the number of permutations of \(1,2,3\), \(\ldots, m\) with \(k\) ascents. Note how \(\pi_{4,2}=11=2(4)+\) \(3(1)=(4-2) \pi_{3,1}+(2+1) \pi_{3,2} .\) How is \(\pi_{m, k}\) related to \(\pi_{m-1, k-1}\) and \(\pi_{m-1, k}\) ? c) If a permutation of \(1,2,3,4,5,6,7\) has four ascents, how many descents does it have?

Write each of the following integers in two's complement representation. Here the results are eight-bit patterns. a) 15 b) \(-15\) c) 100 d) \(-65\) e) 127 f) \(-128\)

Give a recursive definition for the set of all a) positive even integers b) nonnegative even integers

a) How many positive integers can we express as a product of nine primes (repetitions allowed and order not relevant) where the primes may be chosen from \(\\{2,3,5,7,11\\}\) ? b) How many of the positive integers in part (a) have at least one occurrence of each of the five primes?

Write a computer program (or develop an algorithm) to convert a positive integer in base 10 to base \(b\), where \(2 \leq b \leq 9\).
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