To find the prime factorization of 714, we need to break it down into its prime factors step-by-step. Here is how we can do that:
1. Divide by the smallest prime number (2):
714 is even, so it is divisible by 2.
[tex]\[
714 \div 2 = 357
\][/tex]
So, one of the prime factors is 2.
2. Divide the quotient (357) by the next smallest prime number:
Check if 357 is divisible by 2. It isn't, as it is odd.
Next, we check if 357 is divisible by 3. Sum the digits of 357 (3 + 5 + 7 = 15), and since 15 is divisible by 3, so is 357.
[tex]\[
357 \div 3 = 119
\][/tex]
Therefore, another prime factor is 3.
3. Divide the next quotient (119) by the next smallest prime number:
119 is not divisible by 2 or 3 because it doesn’t meet the criteria for those divisions. Now, let's check:
- Is 119 divisible by 5? No, since its last digit is neither 0 nor 5.
- Is 119 divisible by 7? Sum the digits of 119 (1 + 1 + 9 = 11), and let's check:
[tex]\[
119 \div 7 = 17
\][/tex]
So, another prime factor is 7.
4. Finally, check if 17 is a prime number:
17 cannot be divided by any number other than 1 and itself. Thus, 17 is a prime number.
Combining all the factors together, the prime factorization of 714 is:
[tex]\[
714 = 2 \cdot 3 \cdot 7 \cdot 17
\][/tex]
So the correct choice is:
[tex]\[ \boxed{K. \ 2 \cdot 3 \cdot 7 \cdot 17} \][/tex]
