The HCF of two consecutive even numbers is ____.

Question

Infinity Learn · Accepted Answer

Concept- Two consecutive even numbers would have HCF that is equal to 2. Let “a” is to be a factor of “b” if b can be completely divided by leaving no remainder.
As the name itself suggests, HCF or the Highest Common Factor is the largest number which is a common factor to two or more numbers.
Here, we have to find, in a generalized manner, the HCF of two consecutive even numbers.
Let us assume our two consecutive even numbers to be “2n” and “2n+2” where n=0,1,2,3,4…
Now, just by basic observation, we know that

…(1)
And,

.

…(2)
From (1) and (2), it is clearly visible that both the numbers have been formed by multiplying a different number with a 2, i.e., 2 is a common factor of both (2n) and (2n+2).
Other than two, 1 will also be a common factor since it is a common factor of any two numbers considered; But as 1 is less than 2, it cannot be the Highest Common Factor which we are trying to obtain as an answer to the question.
Taking a particular example, let n = 2,
Then, 2n = 4 and (2n+2) = 6.
4 and 6 are two consecutive even numbers and their factors are:
Of 4 are 1,2,4
Of 6 are 1,2,3,6
Again, let n = 20
Then, 2n = 40 and (2n+2) = 42
40 and 42 are also two consecutive even numbers and their factors are:
Of 40 are 1,2,4,5,8,10,20,40
Of 42 are 1,2,3,6,7,14,21,42.
So basically, we can observe that after 2, there are no common factors in the case of two consecutive even numbers i.e., 2 is the HCF of two consecutive even numbers. Hence, the Correct Answer is 2

The HCF of two consecutive even numbers is ____.

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