Find the greatest common factor of 116 x and 128.

Options

A. 2
B. 4
C. 32
D. 116x

Answer

To find the greatest common factor (GCF) of 116x and 128, we will first find the prime factorization of both numbers.

### Step 1: Prime Factorization of 116
1. Start by dividing 116 by the smallest prime number, which is 2:
\[ 116 \div 2 = 58 \]
2. Divide 58 by 2 again:
\[ 58 \div 2 = 29 \]
3. Now, 29 is a prime number and cannot be divided further by any prime number other than itself. Thus, we stop here.

The prime factorization of 116 is:
\[ 116 = 2^2 \times 29 \]

### Step 2: Prime Factorization of 128
1. Start by dividing 128 by 2:
\[ 128 \div 2 = 64 \]
2. Divide 64 by 2:
\[ 64 \div 2 = 32 \]
3. Divide 32 by 2:
\[ 32 \div 2 = 16 \]
4. Divide 16 by 2:
\[ 16 \div 2 = 8 \]
5. Divide 8 by 2:
\[ 8 \div 2 = 4 \]
6. Divide 4 by 2:
\[ 4 \div 2 = 2 \]
7. Finally, divide 2 by 2:
\[ 2 \div 2 = 1 \]

The prime factorization of 128 is:
\[ 128 = 2^7 \]

### Step 3: Finding the GCF
Now that we have the prime factorizations:
- \(116 = 2^2 \times 29\)
- \(128 = 2^7\)

To find the GCF, we take the lowest power of the common prime factors:
- The common prime factor is \(2\).
- The lowest power of \(2\) in both factorizations is \(2^2\).

Thus, the GCF is:
\[ GCF = 2^2 = 4 \]

### Final Answer
The greatest common factor of \(116x\) and \(128\) is \(4\).

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