Find the prime factorization of 63
The Correct Answer and Explanation is :
The prime factorization of 63 is (3^2 \times 7).
Explanation:
To find the prime factorization of a number, we need to express it as a product of prime numbers. Prime numbers are natural numbers greater than 1 that have no positive divisors other than 1 and themselves. The prime numbers relevant to our factorization process include 2, 3, 5, 7, 11, and so on.
- Dividing by the smallest prime: We start by dividing 63 by the smallest prime number, which is 2. Since 63 is odd, it is not divisible by 2. Next, we try the next smallest prime number, which is 3.
- Dividing by 3: We divide 63 by 3:
[
63 \div 3 = 21
]
Since 3 is a prime number, we note it down. - Continuing with 21: Now we take the quotient, 21, and factor it further. We check if 21 is divisible by 3 again:
[
21 \div 3 = 7
]
Again, since 3 is prime, we note it down. Now we have (63 = 3 \times 21 = 3 \times (3 \times 7)). - Identifying the next factor: The quotient now is 7, which is itself a prime number. We can no longer factor it further, and thus we stop here.
- Writing the prime factorization: Collecting all our findings, we see that 63 can be expressed as:
[
63 = 3 \times 3 \times 7 = 3^2 \times 7
] - Conclusion: The prime factorization of 63 is (3^2 \times 7). This breakdown helps in various applications, such as simplifying fractions, finding the greatest common divisor (GCD), and understanding the properties of the number in various mathematical contexts. Prime factorization is a fundamental concept in number theory and serves as the building block for more complex mathematical concepts.