Chapter 6, Problem 1PCT

Find the GCF of 10x³y⁴z, 15x⁵y and 25x²y⁷z³.

To determine

The greatest common factor of the expressions $10x^3{y}^{4}z,15x^{5}y$ and $25x^2{y}^7{z}^3$.

Answer to Problem 1PCT

The greatest common factor of the expressions $10x^3{y}^{4}z,15x^{5}y$ and $25x^2{y}^7{z}^3$ is $5x^{2}y$.

Explanation of Solution

Procedure used:

Greatest common factor:

“Step 1: Find the GCF of the coefficients of each variable expression.

Step 2: For each variable factor common to all the expressions, determine the smallest exponent to which the variable factor is raised.

Step 3: Find the product of the common prime factors found in Steps 1 and 2. This expression is the GCF”.

Calculation:

In order to find the GCF of the expressions $10x^3{y}^{4}z,15x^{5}y$ and $25x^2{y}^7{z}^3$, first find the GCF of the coefficients in the expression.

The coefficients in the expressions, $10x^3{y}^{4}z,15x^{5}y$ and $25x^2{y}^7{z}^3$ are 10, 15 and 25, respectively.

The common factor in the numbers 10, 15 and 25 is 5.

Thus, the GCF of the numbers 10, 15 and 25 is 5.

Now, determine the smallest exponent that is common in the monomials $x^3{y}^{4}z,x^{5}y$ and $x^2{y}^7{z}^3$.

Write the monomials $x^3,x^5$ and $x^2$ as the product of factors of x as below.

$\begin{array}{c}{x}^3=x\cdot x\cdot x\\ x^5=x\cdot x\cdot x\cdot x\cdot x\\ x^2=x\cdot x\end{array}$

Note that, the monomials $x^3,x^5$ and $x^2$ contains $x^2$ in common. That is, the GCF is $x^2$.

Since the variable y is common to $y^4,y$ and $y^7$, the GCF is the smallest power of y that is, y.

Note that, z is common in the first and last term but the second term does not contain z.

That is, the GCF does not contain z.

By step 3 of above procedure, the GCF is $5x^{2}y$.

Thus, the GCF of the expressions $10x^3{y}^{4}z,15x^{5}y$ and $25x^2{y}^7{z}^3$ is $5x^{2}y$.