Question

G.C.D. of ${{\left( a+b-c \right)}^{6}}$ and ${{\left( a+b-c \right)}^{4}}$ is
(a) ${{\left( a+b-c \right)}^{6}}$
(b) ${{\left( a+b-c \right)}^{10}}$
(c) ${{\left( a+b-c \right)}^{2}}$
(d) ${{\left( a+b-c \right)}^{4}}$

Answer 1

Hint: First, before proceeding for this, we must know that the G.C.D. is the greatest common divisor which is defined for the two integers as the largest positive integer that divides each of the integers. Then, by using the above condition, we can find the GCD for the polynomial terms which are ${{\left( a+b-c \right)}^{6}}$ and ${{\left( a+b-c \right)}^{4}}$. Then, by dividing the above terms with the polynomial (a+b-c), we can divide 4 times due to which the second polynomial will be fully divided and further division is not possible which gives the final result.

Complete step by step answer:
In this question, we are supposed to find the .C.D. of ${{\left( a+b-c \right)}^{6}}$ and ${{\left( a+b-c \right)}^{4}}$.
So, before proceeding for this, we must know that the G.C.D. is the greatest common divisor which is defined for the two integers as the largest positive integer that divides each of the integers.
Now, to understand it more carefully, let us take an example as x and y are the two variables with values 8 and 12 respectively.
So, we can clearly see that 4 is the common table in which these two get divided and further there is no number left to divide the remaining of these numbers which gives 4 as the GCD of 8 and 12.
Now, by using the above condition, we can find the GCD for the polynomial terms which are ${{\left( a+b-c \right)}^{6}}$ and ${{\left( a+b-c \right)}^{4}}$.
Then, by dividing the above terms with the polynomial (a+b-c), we can divide 4 times due to which the second polynomial will be fully divided and further division is not possible.
So, the polynomial (a+b-c) with power 4 is the common divisor from both the given polynomials.
So, ${{\left( a+b-c \right)}^{4}}$ is the GCD of the numbers ${{\left( a+b-c \right)}^{6}}$ and ${{\left( a+b-c \right)}^{4}}$.
Hence, option (d) is correct.

Note:
Now, to solve these types of questions we must be careful between the two terms which are LCM and GCD as most of the times these get mixed. So, let us understand with an example as 8 and 12 are two numbers, their GCD is 4 as it is the only common table between them but LCM is the number which we get after multiplication of the remaining terms with the common term as for this case 24 is the LCM. So, be careful while calculating LCM and GCD.