What is the greatest common factor of [tex]$4 k, 18 k^4$[/tex], and [tex]$12$[/tex]?
Answer (2)
The greatest common factor (GCF) of the terms 4 k , 18 k 4 , and 12 is 2 , found by determining the GCF of the coefficients since the variable k does not appear in all terms. We achieved this by finding the prime factorizations and identifying the common factors. Thus, the final answer is 2 .
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Find the prime factorization of the coefficients: 4 = 2 2 , 18 = 2 ⋅ 3 2 , 12 = 2 2 ⋅ 3 .
Identify the common prime factors and their lowest powers: The GCF of the coefficients is 2 1 = 2 .
Check if the variable k is a common factor: Since 12 does not have k , it is not a common factor.
The greatest common factor of 4 k , 18 k 4 , and 12 is 2 .
Explanation
Understanding the Problem We are asked to find the greatest common factor (GCF) of 4 k , 18 k 4 , and 12 . The GCF is the largest factor that divides all the given terms.
Prime Factorization of Coefficients First, let's find the prime factorization of the coefficients of each term:
4 = 2 2 18 = 2 ⋅ 3 2 12 = 2 2 ⋅ 3
Finding GCF of Coefficients Now, we identify the common prime factors among the coefficients. The only common prime factor is 2 . The lowest power of 2 that appears in all the factorizations is 2 1 = 2 . Therefore, the greatest common factor of the coefficients is 2 .
Analyzing the Variable k Next, we consider the variable k . The powers of k in the terms are k 1 and k 4 . However, the term 12 does not contain k at all. Therefore, k is not a common factor of all three terms.
Combining Results Finally, we combine the GCF of the coefficients and the variable part. Since k is not a common factor, the greatest common factor of the entire expression is simply the GCF of the coefficients, which is 2 .
Final Answer Therefore, the greatest common factor of 4 k , 18 k 4 , and 12 is 2 .
Examples
Understanding the greatest common factor (GCF) is useful in many real-life situations. For example, suppose you have three different lengths of fabric: 4 k inches, 18 k 4 inches, and 12 inches. You want to cut the fabric into the longest possible pieces such that all the pieces are of equal length and no fabric is wasted. The GCF of the lengths will give you the length of the longest possible pieces. In this case, the GCF is 2, so you can cut the fabric into pieces of 2 inches each.
