Answer :
Sure, let's find the prime factorization for each of the numbers step-by-step.
### Part (a) Prime Factorization of 16:
1. Identify if 16 is divisible by the smallest prime number (2):
- Yes, because 16 is an even number.
2. Continue dividing by 2:
- [tex]\(16 \div 2 = 8\)[/tex]
- [tex]\(8 \div 2 = 4\)[/tex]
- [tex]\(4 \div 2 = 2\)[/tex]
- [tex]\(2 \div 2 = 1\)[/tex]
3. Since 2 is the only prime factor:
- [tex]\(16 = 2^4\)[/tex]
Therefore, the prime factorization of 16 is [tex]\(2^4\)[/tex].
### Part (b) Prime Factorization of 72:
1. Identify if 72 is divisible by the smallest prime number (2):
- Yes, because 72 is an even number.
2. Continue dividing by 2:
- [tex]\(72 \div 2 = 36\)[/tex]
- [tex]\(36 \div 2 = 18\)[/tex]
- [tex]\(18 \div 2 = 9\)[/tex]
3. At this point, 9 is no longer divisible by 2, so we proceed to the next prime number (3):
- [tex]\(9 \div 3 = 3\)[/tex]
- [tex]\(3 \div 3 = 1\)[/tex]
4. Summarizing the steps, we get:
- [tex]\(72 = 2^3 \times 3^2\)[/tex]
Therefore, the prime factorization of 72 is [tex]\(2^3 \times 3^2\)[/tex].
In conclusion:
- The prime factorization of 16 is [tex]\(2^4\)[/tex].
- The prime factorization of 72 is [tex]\(2^3 \times 3^2\)[/tex].
### Part (a) Prime Factorization of 16:
1. Identify if 16 is divisible by the smallest prime number (2):
- Yes, because 16 is an even number.
2. Continue dividing by 2:
- [tex]\(16 \div 2 = 8\)[/tex]
- [tex]\(8 \div 2 = 4\)[/tex]
- [tex]\(4 \div 2 = 2\)[/tex]
- [tex]\(2 \div 2 = 1\)[/tex]
3. Since 2 is the only prime factor:
- [tex]\(16 = 2^4\)[/tex]
Therefore, the prime factorization of 16 is [tex]\(2^4\)[/tex].
### Part (b) Prime Factorization of 72:
1. Identify if 72 is divisible by the smallest prime number (2):
- Yes, because 72 is an even number.
2. Continue dividing by 2:
- [tex]\(72 \div 2 = 36\)[/tex]
- [tex]\(36 \div 2 = 18\)[/tex]
- [tex]\(18 \div 2 = 9\)[/tex]
3. At this point, 9 is no longer divisible by 2, so we proceed to the next prime number (3):
- [tex]\(9 \div 3 = 3\)[/tex]
- [tex]\(3 \div 3 = 1\)[/tex]
4. Summarizing the steps, we get:
- [tex]\(72 = 2^3 \times 3^2\)[/tex]
Therefore, the prime factorization of 72 is [tex]\(2^3 \times 3^2\)[/tex].
In conclusion:
- The prime factorization of 16 is [tex]\(2^4\)[/tex].
- The prime factorization of 72 is [tex]\(2^3 \times 3^2\)[/tex].
