Let's identify the prime factorization of the number 252 by breaking it down step by step:
1. Initial division by the smallest prime (2):
- [tex]\( 252 \div 2 = 126 \)[/tex]
- We get our first factor [tex]\( 2 \)[/tex].
2. Continue division by 2:
- [tex]\( 126 \div 2 = 63 \)[/tex]
- We get another factor [tex]\( 2 \)[/tex].
At this point, we cannot divide further by 2 since 63 is not divisible by 2.
3. Division by the next smallest prime (3):
- [tex]\( 63 \div 3 = 21 \)[/tex]
- We get a factor [tex]\( 3 \)[/tex].
4. Continue division by 3:
- [tex]\( 21 \div 3 = 7 \)[/tex]
- We get another factor [tex]\( 3 \)[/tex].
Now, the remaining number is 7, which is already a prime number.
So, the complete prime factorization of 252 is:
[tex]\[ 2 \times 2 \times 3 \times 3 \times 7 \][/tex]
Now, let's compare this prime factorization with the given options:
- Option A: [tex]\( 2 \times 2 \times 3 \times 3 \times 7 \)[/tex]
- Option B: [tex]\( 2 \times 2 \times 2 \times 3 \times 7 \)[/tex]
- Option C: [tex]\( 3 \times 3 \times 3 \times 3 \times 7 \)[/tex]
- Option D: [tex]\( 2 \times 3 \times 3 \times 3 \times 7 \)[/tex]
From the comparisons, we can see that the correct prime factorization matches option A.
Therefore, the correct answer is:
[tex]\[ \boxed{1} \][/tex]
