Question

Find the G.C.D of the numbers ${{2}^{4}}\times {{3}^{5}}\times {{5}^{9}}$, ${{3}^{4}}\times {{5}^{3}}\times {{7}^{8}}$ and ${{2}^{7}}\times {{5}^{2}}\times {{7}^{5}}$?
(a) ${{5}^{3}}$,
(b) ${{5}^{2}}$,
(c) ${{3}^{4}}$,
(d) ${{7}^{2}}$.

Answer 1

Hint: We start solving the problem by recalling the definition of G.C.D (Greatest Common divisor) and the procedure of finding it. We then find the common factors that are present in all the given numbers and check that it satisfies all the properties to become a G.C.D (Greatest Common divisor). If it satisfies all the properties, then we declare it as the answer.

Complete step by step answer:
According to the problem, we need to find the G.C.D (Greatest Common divisor) of the numbers ${{2}^{4}}\times {{3}^{5}}\times {{5}^{9}}$, ${{3}^{4}}\times {{5}^{3}}\times {{7}^{8}}$ and ${{2}^{7}}\times {{5}^{2}}\times {{7}^{5}}$.
Before finding the G.C.D (Greatest Common divisor), we recall the definition of it.
We know that the G.C.D (Greatest Common divisor) of two or more integers (not zero) is defined as the largest positive number that can divide all the given integers. We find G.C.D (Greatest Common divisor) by first factoring all the integers and then take the multiplication of the common factors that we obtain from the results of factorization.
The numbers that we are given in the problem are already factored. We need not factorize those numbers again.
We can see that the factor ${{5}^{2}}$ is present in all the given three numbers. So, this is the common factor for all the three given numbers.
We know that 5 is a positive number and the square of a positive number is always positive. So, ${{5}^{2}}$ divides all the three given numbers and is positive and no number larger than ${{5}^{2}}$ divides the three numbers. This makes ${{5}^{2}}$ as the G.C.D (Greatest Common divisor) of the numbers ${{2}^{4}}\times {{3}^{5}}\times {{5}^{9}}$, ${{3}^{4}}\times {{5}^{3}}\times {{7}^{8}}$ and ${{2}^{7}}\times {{5}^{2}}\times {{7}^{5}}$.
We have found the G.C.D (Greatest Common divisor) of the numbers ${{2}^{4}}\times {{3}^{5}}\times {{5}^{9}}$, ${{3}^{4}}\times {{5}^{3}}\times {{7}^{8}}$ and ${{2}^{7}}\times {{5}^{2}}\times {{7}^{5}}$ as ${{5}^{2}}$.
The correct option for the given problem is (b).
Note:
We should know that any of the given numbers should not be zero. We should make sure that the given numbers are integers. We should know that the G.C.D of two or more integers should always be positive and largest. We should not make mistakes while finding the common factor of all the numbers given. Even if we have negative integers present in the problem, our G.C.D will be a positive integer.