Question

Let \(m\) be a positive integer. Find the greatest common divisor of \(m\) and \(m+1\).

Short answer

Answer: The greatest common divisor (GCD) of two consecutive integers is always 1.

Step-by-step solution

Understanding the properties of consecutive integers

Consecutive integers are integers that follow one another in a sequence, such as \(m\) and \(m+1\). Consecutive integers are always co-prime, meaning they have no common factors other than 1. This is because if a number divides both \(m\) and \(m+1\), it must also divide their difference which is 1. The only positive integer that divides 1 is itself; thus, consecutive integers are always co-prime.

Finding the greatest common divisor of consecutive integers

Since consecutive integers are co-prime, the greatest common divisor of two consecutive integers is always 1. Therefore, for any positive integer \(m\), the GCD of \(m\) and \(m+1\) is equal to 1.

Presenting the final solution

In conclusion, the greatest common divisor of any positive integer \(m\) and its consecutive integer \(m+1\) is always 1.

Key concepts

Consecutive Integers

Consecutive integers are numbers that come one after another in order without any gaps. In mathematical terms, if you have an integer represented by the variable 'm', the next consecutive integer is simply 'm+1'. This concept is fundamental in number theory because these sequential numbers have unique properties, especially when it comes to their divisibility.

For instance, no two consecutive integers share any common divisors other than 1. If someone claims that a number 'd' divides both 'm' and 'm+1', then by definition, 'd' would have to divide their difference, which is 'm+1' minus 'm', equaling 1. But the only number that divides 1 is 1 itself, which leads us to the realization that consecutive numbers are always co-prime – a term we will explore next.

Co-prime

Two or more numbers are said to be co-prime (also known as relatively prime) if their only common positive divisor is 1. This means they don't have to be prime numbers themselves; for example, 8 and 9 are not prime numbers, but they are co-prime to each other. In our exercise, we looked at consecutive integers, such as 'm' and 'm+1', and highlighted that by their inherent sequential nature, they must be co-prime.

Understanding the concept of co-prime numbers can be essential when simplifying fractions, finding least common multiples, or like in our exercise, determining the greatest common divisor (GCD). Remember, the GCD of co-prime numbers will always be 1 because they do not share any other divisors. This is a crucial property that simplifies many calculations in number theory.

Number Theory

Number theory is a branch of pure mathematics that deals with the deeper properties of integers. It often involves the study of prime numbers, GCD, and the relationships between numbers as we've seen with consecutive integers being co-prime. One of the cornerstones of number theory is Euclid's algorithm for finding the greatest common divisor of two numbers. This algorithm, also reflected in our exercise solution, emphasizes the importance of understanding the divisions and common factors between integers.

It's exciting to delve into number theory because it isn't just about performing calculations; it's about exploring patterns, making conjectures, and discovering the inherent beauty mathematics has to offer. Knowing the behavior of consecutive integers and co-prime numbers enables mathematicians to solve problems elegantly, just as we found the GCD of 'm' and 'm+1' to be 1. Such elegant solutions are what make number theory both fascinating and broadly applicable in the realm of mathematics.