To find the greatest common factor (GCF) of 140 and 84, we will use the prime factorization method. Let's break down each number into its prime factors.
First, let's factorize 140:
140 can be divided by 2:
[tex]\[ 140 \div 2 = 70 \][/tex]
70 can also be divided by 2:
[tex]\[ 70 \div 2 = 35 \][/tex]
35 is divisible by 5:
[tex]\[ 35 \div 5 = 7 \][/tex]
7 is a prime number. So, the prime factorization of 140 is:
[tex]\[ 140 = 2^2 \times 5 \times 7 \][/tex]
Next, let's factorize 84:
84 can be divided by 2:
[tex]\[ 84 \div 2 = 42 \][/tex]
42 can also be divided by 2:
[tex]\[ 42 \div 2 = 21 \][/tex]
21 is divisible by 3:
[tex]\[ 21 \div 3 = 7 \][/tex]
7 is a prime number. So, the prime factorization of 84 is:
[tex]\[ 84 = 2^2 \times 3 \times 7 \][/tex]
Now, let's write the prime factorization for each number:
[tex]\[
\begin{array}{l}
140 = 2^2 \times 5 \times 7 \\
84 = 2^2 \times 3 \times 7
\end{array}
\][/tex]
Next, identify the common prime factors and their respective smallest powers:
- The prime factor 2 appears to the power of 2 in both factorizations.
- The prime factor 7 appears to the power of 1 in both factorizations.
Now, we multiply these common prime factors with their smallest powers:
[tex]\[ \text{GCF} = 2^2 \times 7^1 = 4 \times 7 = 28 \][/tex]
So, the greatest common factor of 140 and 84 is:
[tex]\[ 28 \][/tex]
