Question

How do you find the greatest common factor of $35{n^2}m,\,\,21{m^2}n$?

Answer 1

Hint:First we should know what is the greatest common factor. It is the HCF of given numbers. In this question we have to find the HCF of two values by prime factorization method. It was for the constant part. For the variable part we have to take the largest power of the variable common to both.

Complete step by step answer:
In the above question,
To find the greatest common factor first we have to find the GCF, or the largest number that goes into both, of $35\,\,and\,\,21$.
Now, let’s do the prime factorization of $35\,\,and\,\,21$.
$ \Rightarrow 35$ has factors $1\,,\,5\,,\,7\,,\,35.$
$ \Rightarrow 21$ has factors $1\,,\,3\,,\,7\,,\,21$.
The greatest (or largest) number common to both lists is $7$.
Next, we have to find the GCF of the variables. To find this, we have to take the largest power of variable common to both.
The largest power of m common to both terms is m. The largest power of n common to both terms is n.
Therefore, the GCF is $7mn$.

Note:In this question if the power of m is raised to two in the first term and raised to three in the second term, then in the final answer there will be ${m^2}$ not m. Similar thing applies to n also. Generally, If we have to find out the GCF of two numbers, we will first list the prime factors of each number. The multiple of common factors of both the numbers results in GCF. But, if there are no common prime factors, the greatest common factor is 1.