Identify the greatest common factor of 40 and [tex]$20 z$[/tex].

Find the prime factorization of 40: 40 = 2 3 × 5 .
Find the prime factorization of 20 z : 20 z = 2 2 × 5 × z .
Identify the common factors: 2 2 and 5 .
Multiply the common factors to find the GCF: 2 2 × 5 = 20 ​ .

Explanation

Problem Analysis We are asked to find the greatest common factor (GCF) of 40 and 20 z . The GCF is the largest factor that both terms share.

Prime Factorization of 40 First, let's find the prime factorization of 40. We have 40 = 2 × 2 × 2 × 5 = 2 3 × 5 .

Prime Factorization of 20z Next, let's find the prime factorization of 20 z . We have 20 z = 2 × 2 × 5 × z = 2 2 × 5 × z .

Identifying Common Factors Now, we identify the common factors between 2 3 × 5 and 2 2 × 5 × z . Both terms have 2 2 and 5 as factors.

Calculating the GCF To find the GCF, we multiply the common factors: 2 2 × 5 = 4 × 5 = 20 . Since z is not a common factor between 40 and 20 z , it is not included in the GCF.

Final Answer Therefore, the greatest common factor of 40 and 20 z is 20 ​ .


Examples
Understanding the greatest common factor is useful in simplifying fractions. For example, if you have a fraction like 20 z 40 ​ , you can simplify it by dividing both the numerator and the denominator by their GCF, which is 20. This simplifies the fraction to z 2 ​ , making it easier to work with. This concept is also crucial in algebra when factoring expressions.

Answered by GinnyAnswer | 2025-07-08

The greatest common factor (GCF) of 40 and 20 z is 20. This is calculated by finding the prime factorizations and identifying the common factors of both numbers. The GCF is obtained by multiplying the lowest powers of these common prime factors.
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Answered by Anonymous | 2025-08-15