What is the greatest common factor of [tex]$60 x^4 y$[/tex], [tex]$45 x^3 y^3$[/tex], and [tex]$75 x^3 y$[/tex]?
A. [tex]$5 x y$[/tex]
B. [tex]$15 x^3 y$[/tex]
C. [tex]$45 x^3 y^5$[/tex]
D. [tex]$75 x^5 y^7$[/tex]
Answer (2)
The greatest common factor of 60 x 4 y , 45 x 3 y 3 , and 75 x 3 y is 15 x 3 y . The calculations involve finding the GCF of the coefficients and the lowest powers of each variable. The correct multiple choice option is B: 15 x 3 y .
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Find the greatest common factor (GCF) of the coefficients 60, 45, and 75, which is 15.
Identify the lowest power of x in the terms x 4 , x 3 , and x 3 , which is x 3 .
Identify the lowest power of y in the terms y , y 3 , and y , which is y .
Combine these to form the GCF of the entire expression: 15 x 3 y .
Explanation
Problem Analysis We are asked to find the greatest common factor (GCF) of 60 x 4 y , 45 x 3 y 3 , and 75 x 3 y . The GCF is the largest expression that divides evenly into all three terms.
GCF of Coefficients First, let's find the GCF of the coefficients: 60, 45, and 75. The prime factorization of each number is:
60 = 2 2 ⋅ 3 ⋅ 5 45 = 3 2 ⋅ 5 75 = 3 ⋅ 5 2
The common factors are 3 and 5. The lowest power of 3 is 3 1 and the lowest power of 5 is 5 1 . Therefore, the GCF of the coefficients is 3 ⋅ 5 = 15 .
GCF of x terms Next, let's find the lowest power of x present in all three terms. The terms are x 4 , x 3 , and x 3 . The lowest power is x 3 .
GCF of y terms Now, let's find the lowest power of y present in all three terms. The terms are y 1 , y 3 , and y 1 . The lowest power is y 1 or simply y .
Combining the Factors Finally, we multiply the GCF of the coefficients with the lowest powers of x and y to obtain the GCF of the three terms: 15 ⋅ x 3 ⋅ y = 15 x 3 y .
Final Answer Therefore, the greatest common factor of 60 x 4 y , 45 x 3 y 3 , and 75 x 3 y is 15 x 3 y .
Examples
Understanding the greatest common factor (GCF) is incredibly useful in various real-life scenarios. For instance, imagine you're tiling a rectangular floor. You want to use the largest square tiles possible to cover the floor without cutting any tiles. If your floor is 60 inches by 45 inches, the side length of the largest square tile you can use is the GCF of 60 and 45, which is 15 inches. This ensures you cover the floor perfectly with whole tiles, saving time and reducing waste. Similarly, GCF helps in dividing tasks or resources into the largest equal groups possible, optimizing efficiency and fairness.
