Answer :
- Divide 126 by the smallest prime number, 2, to get $126 = 2 \times 63$.
- Divide 63 by the next smallest prime number, 3, to get $63 = 3 \times 21$.
- Divide 21 by 3 to get $21 = 3 \times 7$.
- Since 7 is a prime number, the prime factorization of 126 is $2 \times 3^2 \times 7$, so the answer is $\boxed{2 \times 3^2 \times 7}$.
### Explanation
1. Problem Analysis
We are asked to express 126 as a product of its prime factors. The provided table has an error, since 126/2 = 63, not 13.
2. Dividing by the Smallest Prime
We start by dividing 126 by the smallest prime number, 2: $126 = 2 \times 63$.
3. Finding Prime Factors of 63
Now, find the prime factors of 63. Since 63 is not divisible by 2, try the next smallest prime number, 3: $63 = 3 \times 21$.
4. Finding Prime Factors of 21
Next, find the prime factors of 21. Again, divide by 3: $21 = 3 \times 7$.
5. Prime Factorization of 7
Finally, 7 is a prime number, so the prime factorization is complete: $7 = 7 \times 1$.
6. Combining Prime Factors
Combine all the prime factors: $126 = 2 \times 3 \times 3 \times 7 = 2 \times 3^2 \times 7$.
7. Final Answer
Therefore, the prime factorization of 126 is $2 \times 3^2 \times 7$.
### Examples
Prime factorization is used in cryptography to ensure secure data transmission. For example, the security of the RSA algorithm relies on the fact that it is difficult to factor large numbers into their prime factors. In real life, this is used to protect credit card numbers and other sensitive information online.
- Divide 63 by the next smallest prime number, 3, to get $63 = 3 \times 21$.
- Divide 21 by 3 to get $21 = 3 \times 7$.
- Since 7 is a prime number, the prime factorization of 126 is $2 \times 3^2 \times 7$, so the answer is $\boxed{2 \times 3^2 \times 7}$.
### Explanation
1. Problem Analysis
We are asked to express 126 as a product of its prime factors. The provided table has an error, since 126/2 = 63, not 13.
2. Dividing by the Smallest Prime
We start by dividing 126 by the smallest prime number, 2: $126 = 2 \times 63$.
3. Finding Prime Factors of 63
Now, find the prime factors of 63. Since 63 is not divisible by 2, try the next smallest prime number, 3: $63 = 3 \times 21$.
4. Finding Prime Factors of 21
Next, find the prime factors of 21. Again, divide by 3: $21 = 3 \times 7$.
5. Prime Factorization of 7
Finally, 7 is a prime number, so the prime factorization is complete: $7 = 7 \times 1$.
6. Combining Prime Factors
Combine all the prime factors: $126 = 2 \times 3 \times 3 \times 7 = 2 \times 3^2 \times 7$.
7. Final Answer
Therefore, the prime factorization of 126 is $2 \times 3^2 \times 7$.
### Examples
Prime factorization is used in cryptography to ensure secure data transmission. For example, the security of the RSA algorithm relies on the fact that it is difficult to factor large numbers into their prime factors. In real life, this is used to protect credit card numbers and other sensitive information online.
