Question

Find the HCF of the numbers given. k, 2k, 3k, 4k and 5k, where k is any positive integer.

Answer 1

Hint: In this question find the factors of the given numbers. Then find for the highest common numbers present in all the numbers to find the H.C.F of the numbers.

Complete step-by-step answer:
Given numbers are
k, 2k, 3k, 4k and 5k, where k is any positive integer.
We have to find out the H.C.F of these numbers.
As we know H.C.F of any numbers is the multiplication of common factors.
Now as we see in the given numbers the factors of numbers are
$\left( {1 \times k} \right)$,$\left( {1 \times 2 \times k} \right)$,$\left( {1 \times 3 \times k} \right)$, $\left( {1 \times 4 \times k} \right)$ and$\left( {1 \times 5 \times k} \right)$, and we all know that (1, 2, 3, 4 and 5) are all prime numbers so we cannot factorize further.
A prime number is a number which is a multiple of 1 and itself.
So from the amongst factors the common factor is only $\left( {1 \times k} \right) = k$
So the H.C.F of the given numbers is k.
So this is the required answer.

Note: In this question it is given that k is any positive integer, if we talk about k being negative than what happens? The key point to understand is that the greatest common factor of two negative numbers can never be negative. Let’s understand it with an example. Take any two negative numbers e.g. -12 and -9.
Prime factors of -12 is ${\text{ - 2}} \times {\text{2}} \times {\text{3 or 2}} \times {\text{2}} \times - {\text{3 or - 2}} \times - {\text{2}} \times - {\text{3}}$
Prime factors of -9 is $ - 3 \times 3$
But if we have a look for -12 it has 3 factors so, in order to maintain the negative sign of 12, in its factors either one should be negative or all should be negative.
Now the greatest common factor will be 3 as -3 is also a contender but obviously 3>-3.