Question

Find the HCF of two consecutive even numbers _______.

Answer 1

Hint: We will take some general examples of even consecutive numbers to find the Highest Common Factor of two consecutive even numbers. General even number can be represented as $2n$ and two consecutive even numbers can be represented as $(2n,2n+2)$.

Complete step-by-step answer:
It is given in the question to find Highest Common Factor (HCF) of two consecutive even numbers. Here, the numbers which come one after another on the number line are known to be consecutive numbers. For example, $2,3,4,5$ where $2$ and $3$ are consecutive numbers. Similarly, $3$ and $4$ are consecutive numbers and so on.
Even numbers are the natural numbers which are divisible by $2$ and they can be represented as factors of $2$. So, we can represent even numbers with $2(n)$ where $n$ is any natural number and $2$ is its factor. As we have to find the Highest Common Factor (HCF) of even consecutive numbers, they are: $2,4,6,8,10$ and so on.
We can also represent any two consecutive even numbers as $2$ and $(2n+2)$, where $(2n+2)$ is also an even number next to $2n$. Let first even number be $2n$ then the second consecutive even number will be $(2n+2)$.
Now, $2n$ can be written as $2\times (n)$ and $(2n+2)$ as $2(n+1)$.
On comparing $2n$ and $2(n+1)$, we get two numbers $(n)$ and $(n+1)$ which are probably two consecutive integers and also prime numbers. Basically, prime numbers are the numbers which have only two factors, the first is $1$ and the second is the number itself. For example, $5,7,11$ etc. there is no common factor between $(n)$ and $(n+1)$ other than $1$. So, they are known to be prime numbers.
Therefore, from the above discussion, we get the Highest Common Factor between $2n$ and $(2n+2)$ is $2$. As a result, the Highest Common Factor of any two consecutive numbers is $2$.

Note: As, we have a set of even numbers it is clear that even numbers have $2$ as its factor. Likewise, it is an individual number which is the factor of all even numbers set together. Highest Common Factor means the largest or the greatest factor common to any two or more given natural numbers, it is also known as Greatest Common Divisor (GCD). For example, we have three numbers $14,21$ and $35$ to which the Highest Common Factor is $7$ that can divide all the three numbers listed and the remainder is zero.