To determine the prime factorization of the number 147, let's go through the steps to factorize it into its prime components.
1. Divide by the smallest prime number possible:
- 147 is odd, so it is not divisible by 2.
- The sum of the digits of 147 is [tex]\(1 + 4 + 7 = 12\)[/tex], which is divisible by 3. Hence, 147 is divisible by 3.
- [tex]\(147 ÷ 3 = 49\)[/tex].
2. Factorize the quotient:
- Now we need to factorize 49.
- 49 is not divisible by 2 or 3.
- 49 is [tex]\(7 \times 7\)[/tex], and 7 is a prime number.
So, we have:
- [tex]\(147 = 3 \times 49\)[/tex]
- [tex]\(49 = 7 \times 7\)[/tex]
Therefore, combining these, the prime factorization of 147 is:
[tex]\[ 147 = 3 \times 7 \times 7 \][/tex]
Now, let's compare this factorization with the options provided:
- (A) [tex]\(2 \times 2 \times 3 \times 3 \times 3\)[/tex] – This option does not match our factorization.
- (B) [tex]\(2 \times 3 \times 4 \times 5\)[/tex] – This option does not match our factorization.
- (C) [tex]\(3 \times 7 \times 9\)[/tex] – This option does match numerically but 9 is not a prime factor.
- (D) [tex]\(3 \times 7 \times 7\)[/tex] – This option matches exactly.
Hence, the correct option is:
[tex]\[ \boxed{D} \][/tex]
