Question

Prove that if \(m\) is odd and \(n\) is even, then gcd \((m,\)\\[n)=\operatorname{gcd}(m, n / 2)\\]

Short answer

Using the properties of the gcd and odd-even numbers, it can be proved that if \(m\) is odd and \(n\) is even, then gcd(\(m,n\)) equals gcd(\(m, \frac{n}{2}\)).

Step-by-step solution

Understand the properties of odd, even numbers and gcd

An odd number is one which is not divisible by 2, while an even number is divisible by 2. Greatest Common Divisor (gcd) of two integers \(m\) and \(n\), not both equal to zero, is the largest positive integer that divides both \(m\) and \(n\) without a remainder.

Assume \(m\) and \(n\) are both not zero

Since gcd is defined for two integers not both equal to zero, let’s assume \(m\) and \(n\) are both not zero. Also, \(m\) is odd and \(n\) is even as given in the problem statement.

Define gcd and equality

If \(d\) is gcd(\(m, n\)), then \(d\) divides \(m\) and \(d\) also divides \(n\). Hence, it must divide \(\frac{n}{2}\) as well because \(n\) is even. Also, if \(d'\) is a divisor of both \(m\) and \(\frac{n}{2}\), given that \(m\) is odd and \(\frac{n}{2}\) is half of an even integer, \(2d'\) would also be a divisor of \(n\). Hence, \(d'\) is a divisor of both \(m\) and \(n\). This means \(d = d'\) and hence gcd(\(m, n\)) = gcd(\(m, \frac{n}{2}\)).

Conclusion

From the above steps, we can conclude that if \(m\) is odd and \(n\) is even, then gcd(\(m,n\))=gcd(\(m, \frac{n}{2}\)).

Key concepts

Odd and Even Numbers

In mathematics, distinguishing between odd and even numbers is crucial because it lays the foundation for more complex concepts, such as divisibility and congruence.

• An odd number is a number that cannot be evenly divided by 2. When an odd number is divided by 2, it leaves a remainder of 1. Examples of odd numbers include 1, 3, 5, 7, etc.


• An even number, on the other hand, can be divided by 2 without any remainder. Examples of even numbers are 2, 4, 6, 8, etc.

Understanding these definitions is essential when working with concepts like the Greatest Common Divisor (GCD). When one number is odd and the other is even, it influences how their common divisors behave, particularly when it comes to simplifying or rearranging terms, like halving an even number during calculations.

Divisor

The term divisor is a fundamental concept in number theory.

• A divisor of a number is an integer that can divide the number without leaving a remainder.


• A number might have multiple divisors, including 1 and the number itself; these are known as its trivial divisors.

The Greatest Common Divisor (GCD) of two numbers is the largest number that can divide both of them without leaving a remainder. In the expression gcd(

Since n is even, it is divisible by 2, hence simplifying gcd matters when comparing with solutions like gcd( ). This method of computation brings efficiency when handling large numbers and simplifies steps in proofs.

Number Theory

Number theory is a branch of pure mathematics devoted primarily to the study of integers. It deals with the properties and relationships of numbers, particularly the more complex interactions among integers.

• Prime numbers, divisibility, and modular arithmetic are key topics in number theory that interconnect to solve equations involving integers.


• The concept of GCD is integral to number theory. It is used in various algorithms including those for computing the simplification of fractions, solving Diophantine equations, and designing efficient coding schemes.

In this exercise, number theory principles help us understand the relationship between odd and even numbers with respect to their GCD. Applying divisor properties allows for a clearer proof where the GCD of an odd number and half of an even number is equal to the GCD of the same odd number and the original even number, emphasizing the elegance and utility of number theoretical methods.