Prime factorization of 66 by division method

General
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prime factorization of 66 by division method

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What is the prime factorization of 66 using the division method?

Answer:

To find the prime factorization of 66 using the division method, we will divide the number by the smallest prime numbers, repeatedly and sequentially, until we reach 1. This process helps decompose the number into its prime factors. Let’s go through the step-by-step procedure:

Step-by-Step Prime Factorization of 66

  1. Start with the smallest prime number:
    The smallest prime number is 2. Check if 66 is divisible by 2. Since 66 is even, it is divisible by 2.

    $$ 66 \div 2 = 33 $$

    Thus, we write:

    • 66 = 2 × 33

  2. Proceed to the next smallest prime number:
    Now, 33 is the quotient. We need to determine its smallest prime factor. Since 33 is not even, it is not divisible by 2. Therefore, we move to the next prime number, which is 3.

    $$ 33 \div 3 = 11 $$

    Thus, we continue with:

    • 33 = 3 × 11

  3. Factor the resulting quotient:
    The quotient is now 11, which itself is a prime number. So, we can’t factor it any further.

Final Prime Factorization

Putting it all together, we have divided 66 by the smallest primes sequentially: first by 2, then by 3, and finally by 11. Therefore, the prime factorization of 66 is:

$$ 66 = 2 \times 3 \times 11 $$

In list form:

  • 2 is a prime factor.
  • 3 is a prime factor.
  • 11 is a prime factor.

Summary Table

Step Division Quotient Prime Factor
1 66 ÷ 2 33 2
2 33 ÷ 3 11 3
3 11 ÷ 11 1 11

Explanation of Prime Numbers

The prime numbers used in this factorization are:

  • 2: The smallest and only even prime number.
  • 3: Next smallest prime number.
  • 11: A prime number since it has no divisors other than 1 and itself.

Importance of Prime Factorization

Prime factorization is the foundation for numerous concepts in mathematics, including:

  • Greatest Common Divisor (GCD): Helps in finding the GCD of numbers by comparing prime factors.
  • Least Common Multiple (LCM): Used to determine the LCM by combining prime factors.
  • Cryptography: Prime numbers play a critical role in modern cryptography algorithms.

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