Question

Show that the gcd operator is associative. That is, for all integers \(m, n,\) and \(h,\) we have gcd \((m,\) \(g c d(n, h))=\operatorname{gcd}(\operatorname{gcd}(m, n), h)\).

Short answer

The gcd operation is associative for any integers \(m, n, h\) as proven by showing that the result of the gcd operation doesn't change due to the grouping of numbers. The assumptions satisfy the conditions for being the gcd, hence confirming the equality \(g c d(m, g c d(n, h))\) does indeed equal \(g c d(g c d(m, n), h)\).

Step-by-step solution

Definition and Statement of Property

The Greatest Common Divisor (gcd) of two integers is the largest integer that divides both numbers without leaving a remainder. The associative property states that the way numbers are grouped does not change their gcd. In this context, it means gcd \((m,\) \(g c d(n,h))=g c d(g c d(m, n), h)\). This is the property that needs to be proven for all integers \(m, n,\) and \(h\).

Proving the Left Side of the Equation

The goal is to show that \(d\) is a common divisor of \(m\) and \(g c d(n, h)\), meaning \(d\) divides \(m\) and also divides both \(n\) and \(h\). Given that, \(d\) must be the gcd of \(m\), \(n\), and \(h\). This is represented as \(d | m\) and \(d | n\) and \(d | h\). Therefore, per definition of gcd, \(d\) must be equal or lesser than gcd\((m, g c d(n,h))\).

Proving the Right Side of the Equation

Now, let's prove that \(d\) is also a common divisor of \(g c d(m, n)\) and \(h\), which is represented as \(d | g c d(m, n)\) and \(d | h\). Therefore, per definition of gcd, \(d\) should be equal or lesser than \(g c d(g c d(m, n), h)\). If both these conditions from Step 2 and Step 3 holds true, then it implies that \(g c d(m, g c d(n, h)) = g c d(g c d(m, n), h)\).

Checking Both Sides

Since our assumptions in Step 2 and Step 3 satisfy the conditions for being the gcd and our assumption \(d\) is valid for both sides of the equation, we can therefore conclude that \(g c d(m, g c d(n, h))\) does indeed equal \(g c d(g c d(m, n), h)\). This confirms that the gcd operation is associative for any integers \(m, n, h\).

Key concepts

Greatest Common Divisor (gcd)

The Greatest Common Divisor (gcd), also known as the highest common factor, is a fundamental concept in number theory. It refers to the largest positive integer that evenly divides two or more integers without leaving a remainder. For example, the gcd of 8 and 12 is 4, because 4 is the largest number that can divide both 8 and 12 without a remainder.

Understanding the gcd is essential for many areas of mathematics, including solving equations, simplifying fractions, and analyzing prime factors. In practice, finding the gcd of two numbers can be done using the Euclidean algorithm, which involves a series of division steps to reduce the problem to smaller equivalents until a solution is found. This method is particularly effective and is the basis for many computer algorithms that calculate gcd.

Proof Technique

Proof technique is the method or approach used to demonstrate the truth of a mathematical statement. There are various proof techniques such as direct proof, proof by contradiction, inductive proof, and more. For associativity of the gcd operation, a direct proof approach is effective. This involves showing that, for any choice of integers, the grouping of the numbers when calculating their gcd does not affect the result.

Direct proof generally starts with known definitions and properties and uses logical reasoning to arrive at the required conclusion. The essence of such proofs is to construct a logical and convincing argument that the property in question—here, the associative nature of the gcd operation—holds for all relevant cases. It is also important to address each side of the equation separately, as shown in the steps of the solution, to establish that the property works in both directions.

Mathematical Properties

Mathematical properties are rules that explain how numbers behave and relate to each other. Some of the most fundamental properties include commutative, associative, distributive, and identity properties. Each property has a unique role in simplifying expressions and solving equations.

The associative property, in particular, focuses on the grouping within an expression containing two or more of the same associative operations. For addition and multiplication, this property states that no matter how the numbers are grouped, the result will be the same. For the gcd, establishing the associative property guarantees that the order in which pairs of numbers are evaluated does not affect the final gcd result. This is crucial for complex calculations where the gcd is used in sequences of numbers, ensuring consistency and reliability in mathematical proofs and applications.

By understanding these properties, mathematicians and students can navigate complicated problems more efficiently and with greater clarity.