Find the prime factors of these numbers using the division method.
a. 84 b. 117 c. 333 d. 126 e. 520 f. 99
The correct answer and explanation is:
Prime Factorization using the Division Method:
a. 84
Start by dividing 84 by the smallest prime number, 2, as long as it is divisible:
- 84 ÷ 2 = 42
- 42 ÷ 2 = 21
- 21 ÷ 3 = 7 (7 is prime)
So, the prime factorization of 84 is:
84 = 2 × 2 × 3 × 7
b. 117
Start by dividing 117 by the smallest prime number, 3:
- 117 ÷ 3 = 39
- 39 ÷ 3 = 13 (13 is prime)
So, the prime factorization of 117 is:
117 = 3 × 3 × 13
c. 333
Start by dividing 333 by the smallest prime number, 3:
- 333 ÷ 3 = 111
- 111 ÷ 3 = 37 (37 is prime)
So, the prime factorization of 333 is:
333 = 3 × 3 × 37
d. 126
Start by dividing 126 by the smallest prime number, 2:
- 126 ÷ 2 = 63
- 63 ÷ 3 = 21
- 21 ÷ 3 = 7 (7 is prime)
So, the prime factorization of 126 is:
126 = 2 × 3 × 3 × 7
e. 520
Start by dividing 520 by the smallest prime number, 2:
- 520 ÷ 2 = 260
- 260 ÷ 2 = 130
- 130 ÷ 2 = 65
- 65 ÷ 5 = 13 (13 is prime)
So, the prime factorization of 520 is:
520 = 2 × 2 × 2 × 5 × 13
f. 99
Start by dividing 99 by the smallest prime number, 3:
- 99 ÷ 3 = 33
- 33 ÷ 3 = 11 (11 is prime)
So, the prime factorization of 99 is:
99 = 3 × 3 × 11
Explanation of the Division Method:
The division method is used by successively dividing the given number by the smallest prime numbers (2, 3, 5, 7, etc.) until the quotient is 1. Each divisor used in this process is a prime factor of the original number. For example, when factorizing 84, we begin dividing by 2 since 84 is even, and then continue dividing by 3 and 7 as necessary until all factors are prime. This method ensures that the number is broken down into its prime factors.