Question

Find the greatest common factor of the given monomials.
1. $(3m,6n)$
2.$(ax, - 5ab)$

Answer 1

Hint: Take each monomial and write its factorization. Compare the monomials to find the common factors. Consider the product of these common factors if there are more than one factors which are common. If there is one common factor, then that factor is the greatest common factor
If there are more than one common factors, then the product of these factors is the required answer.

Complete step by step solution: We are given two pairs of monomials.
Our aim is to find the greatest common factor i.e. the GCF of the pairs.
1. Consider$(3m,6n)$
In order to find the greatest common factor, we must first determine the factors of the given monomials.
Let’s factorize them one by one.
3m = 3 × m × 1
Here, 3 is a prime number and the power of m is 1.
Therefore, 3m has only three factors which are 1, 3, and m.
6n = 2 × 3 × n
Thus, the factors of 6n are 2, 3, and n.
Then, the common factor of 3m and 6n is 3.
Hence the greatest common factor of 3m and 6n is 3.
2. Consider$(ax, - 5ab)$
Factorize ax.
ax = a × x
There are only 2 variables in ax, which are a and x and their powers are 1 each.
Therefore, a, x are the prime factors of ax.
Now, factorize -5ab.
Then, we get -5ab = -5 × a × b.
Therefore, the prime factors of -5ab are 1,-5, a, and b.
As we can see, the common factors of the monomials ax, -5ab is a.
Hence the greatest common factor is a.

Note: One should remember that in the case of monomials, there cannot be a factor with a power greater than one. Therefore, there will be no such factors as the GCF. Remember this fact to avoid incorrect multiplication of factors.