Prime factorization of 66

General
1

prime factorization of 66

2

Prime Factorization of 66

Answer:

Prime factorization is the process of expressing a number as a product of its prime numbers. A prime number is a number greater than 1 that has no divisors other than 1 and itself.

To find the prime factorization of 66, we’ll follow these steps:

  1. Identify the Smallest Prime Factor:

    • Start with the smallest prime number, which is 2. Check if 66 is divisible by 2. Since 66 is an even number, it is divisible by 2.
    • Perform the division: (66 \div 2 = 33).

  2. Continue with Next Factor of 33:

    • Now, consider the number 33. Check for the smallest prime number by which 33 can be divided.
    • The next prime number is 3. Check divisibility: (33 \div 3 = 11).

  3. Check Prime Nature of Remaining Number:

    • The remaining number is 11, which is itself a prime number.

  4. Conclude the Process:

    • Thus, the prime factorization process stops as all factors are prime numbers.

The prime factorization of 66 is:

[
66 = 2 \times 3 \times 11
]

To verify, multiply these prime factors together:

[
2 \times 3 = 6 \
6 \times 11 = 66
]

Hence, the prime factorization of 66 is correct.

Additional Explanation and Verification

Each of these numbers is prime:

  • 2: The smallest and the only even prime number.
  • 3: A prime number as its only divisors are 1 and 3.
  • 11: Also prime as it cannot be divided by any other number except 1 and 11.

Moreover, when you’re solving problems or simplifying fractions involving 66, recognizing its prime factors can be incredibly useful, as it allows you to simplify or solve by canceling out common factors with other numbers. Using this prime factorization technique ensures that you’re breaking down numbers into their most fundamental components, providing clear insights into their divisibility and their relationship with other numbers.

By identifying and using the prime factors, you can tackle various number theory problems effectively, such as finding the greatest common divisor (GCD) or the least common multiple (LCM) of numbers—practical applications that stretch across many mathematical and real-world scenarios.