Give the prime factorization of each of the following numbers :
`252`

Options

A. `2 xx 2 xx 3 xx 7`
B. `2 xx 3 xx 7`
C. `2 xx 2 xx 3 xx 3 xx 7`
D. none of these

Answer

To find the prime factorization of the number 252, we will follow these steps:

### Step 1: Start with the number 252
We begin with the number 252 and will divide it by the smallest prime number, which is 2.

### Step 2: Divide by 2
252 is an even number, so we can divide it by 2:
\[ 252 \div 2 = 126 \]
So, we have:
\[ 252 = 2 \times 126 \]

### Step 3: Divide 126 by 2 again
Now, we take 126 and divide it by 2 again:
\[ 126 \div 2 = 63 \]
So, we can update our factorization:
\[ 252 = 2 \times 2 \times 63 \]
or
\[ 252 = 2^2 \times 63 \]

### Step 4: Divide 63 by 3
Next, we take 63 and divide it by the next smallest prime number, which is 3:
\[ 63 \div 3 = 21 \]
So, we can update our factorization:
\[ 252 = 2^2 \times 3 \times 21 \]

### Step 5: Divide 21 by 3 again
Now, we take 21 and divide it by 3 again:
\[ 21 \div 3 = 7 \]
So, we can update our factorization:
\[ 252 = 2^2 \times 3 \times 3 \times 7 \]
or
\[ 252 = 2^2 \times 3^2 \times 7 \]

### Step 6: Finalize the factorization
Now, we have reached the prime numbers:
- The prime factors of 252 are 2, 3, and 7.
- The complete prime factorization of 252 is:
\[ 2^2 \times 3^2 \times 7 \]

### Final Answer:
The prime factorization of 252 is:
\[ 2^2 \times 3^2 \times 7 \]

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