What is the greatest common factor of [tex]$42 a^5 b^3, 35 a^3 b^4$[/tex], and [tex]$42 a b^4 ?$[/tex]
Answer (2)
The greatest common factor (GCF) of the expressions 42 a 5 b 3 , 35 a 3 b 4 , and 42 a b 4 is 7 a b 3 . This is determined by finding the GCF of the coefficients and the lowest powers of the variables present in all three terms. Therefore, the answer is 7 a b 3 .
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Find the greatest common factor (GCF) of the coefficients: GCF ( 42 , 35 , 42 ) = 7 .
Find the lowest power of a : min ( a 5 , a 3 , a 1 ) = a 1 = a .
Find the lowest power of b : min ( b 3 , b 4 , b 4 ) = b 3 .
Multiply the GCF of the coefficients and the lowest powers of a and b : 7 ⋅ a ⋅ b 3 = 7 a b 3 .
Explanation
Problem Analysis We are asked to find the greatest common factor (GCF) of the expressions 42 a 5 b 3 , 35 a 3 b 4 , and 42 a b 4 . The GCF is the largest expression that divides evenly into all given expressions. We will find the GCF of the coefficients and the variables separately.
GCF of Coefficients First, let's find the GCF of the coefficients 42, 35, and 42. We can use prime factorization to find the GCF. The prime factorization of 42 is 2 × 3 × 7 , and the prime factorization of 35 is 5 × 7 . The common factor is 7. Therefore, the GCF of the coefficients is 7.
Lowest Power of a Next, let's find the lowest power of a present in all three terms. The powers of a are a 5 , a 3 , and a 1 . The lowest power is a 1 = a .
Lowest Power of b Now, let's find the lowest power of b present in all three terms. The powers of b are b 3 , b 4 , and b 4 . The lowest power is b 3 .
Combining the Results Finally, we multiply the GCF of the coefficients with the lowest powers of a and b to obtain the GCF of the given expressions. The GCF is 7 × a × b 3 = 7 a b 3 .
Final Answer Therefore, the greatest common factor of 42 a 5 b 3 , 35 a 3 b 4 , and 42 a b 4 is 7 a b 3 .
Examples
Understanding the greatest common factor (GCF) is useful in many real-life situations. For instance, suppose you are tiling a rectangular floor with dimensions 42 a 5 b 3 inches by 35 a 3 b 4 inches, where a and b represent some units of length. If you want to use the largest possible square tiles without cutting any tiles, the side length of the square tile would be the GCF of the dimensions. In this case, the side length of the largest square tile you can use is 7 a b 3 inches. This ensures that you can perfectly cover the floor without needing to cut any tiles, saving time and materials.
