Question

How do you find the GCF of $25{x^3}$, $45{x^4}$ and $65{x^2}$?

Answer 1

Hint: The greatest positive integer that divides equally into all numbers with zero left as the remainder is the greatest common factor (GCF) or the highest common factor (HCF) of a group of whole numbers. There are many ways to find the greatest common factor or the GCF of the numbers.
Here, the terms are in terms of variables. We can use the prime factorisation method here. We write all the prime factors of the given numbers and expand the variables given in exponent form. Then, we note down the common prime factors in all numbers and then we multiply the common factors to get the highest common factor.

Complete step by step solution:
We have the numbers as $25{x^3}$, $45{x^4}$ and $65{x^2}$.
We use the prime factorisation method here to find the greatest common factor of the given numbers.
By the factoring method, we have:
The factorisation of $25{x^3}$ is $25{x^3} = 5 \times 5 \times x \times x$.
The factorisation of $45{x^4}$ is $45{x^4} = 3 \times 3 \times 5 \times x \times x \times x \times x$.
The factorisation of $65{x^2}$ is $65{x^2} = 5 \times 13 \times x \times x$.
Here, we can note down the common factors of $25{x^3}$, $45{x^4}$ and $65{x^2}$ which are $5$, $x$ and $x$.
So, we multiply the common factors $5$, $x$ and $x$ to get the greatest common factor.
Thus, GCF $ = 5 \times x \times x\; = 5{x^2}$

Hence, $5{x^2}$ is the GCF of $25{x^3}$, $45{x^4}$ and $65{x^2}$

Note: Prime factorisation method is just one method to find the greatest common factor. We can also use the factoring method, wherein we write down all the factors (not prime factors like the method above) of the numbers and then we check which of the factors are common and the greatest among those common factors would be GCF or HCF. However, the factorising method will be very lengthy in this case because the variables carry exponents.