Question
You found the prime factorization of the number 72. Explain how you can check your answer.
Answers
Answered by
GPT-4o mini
To check the prime factorization of the number 72, you can follow these steps:
1. **Verify the Prime Factors:**
First, ensure that the factors you found are indeed prime numbers. The prime factorization of 72 is:
\( 72 = 2^3 \times 3^2 \), where both 2 and 3 are prime numbers.
2. **Reconstruct the Original Number:**
Multiply the prime factors together to see if you arrive back at the original number, 72.
You can do this as follows:
\[
2^3 = 2 \times 2 \times 2 = 8
\]
\[
3^2 = 3 \times 3 = 9
\]
Now multiply these two results together:
\[
8 \times 9 = 72
\]
3. **Check the Factorization Process:**
Alternatively, you can examine how the factorization was derived by continuously dividing 72 by prime numbers until you reach 1. Start with 72:
- Divide by 2: \( 72 \div 2 = 36 \) (1st factor of 2)
- Divide by 2 again: \( 36 \div 2 = 18 \) (2nd factor of 2)
- Divide by 2 again: \( 18 \div 2 = 9 \) (3rd factor of 2)
- Now the next prime number is 3. Divide by 3: \( 9 \div 3 = 3 \) (1st factor of 3)
- Divide by 3 again: \( 3 \div 3 = 1 \) (2nd factor of 3)
You now have three 2's and two 3's, confirming that the prime factorization is \( 2^3 \times 3^2 \).
4. **Count and Check:**
Count the total number of factors generated in the factorization. If all calculations have been done correctly, you should have matched the total to the original number upon multiplication.
If both methods confirm that you return to the original number of 72, you can confidently say that your prime factorization is correct.
1. **Verify the Prime Factors:**
First, ensure that the factors you found are indeed prime numbers. The prime factorization of 72 is:
\( 72 = 2^3 \times 3^2 \), where both 2 and 3 are prime numbers.
2. **Reconstruct the Original Number:**
Multiply the prime factors together to see if you arrive back at the original number, 72.
You can do this as follows:
\[
2^3 = 2 \times 2 \times 2 = 8
\]
\[
3^2 = 3 \times 3 = 9
\]
Now multiply these two results together:
\[
8 \times 9 = 72
\]
3. **Check the Factorization Process:**
Alternatively, you can examine how the factorization was derived by continuously dividing 72 by prime numbers until you reach 1. Start with 72:
- Divide by 2: \( 72 \div 2 = 36 \) (1st factor of 2)
- Divide by 2 again: \( 36 \div 2 = 18 \) (2nd factor of 2)
- Divide by 2 again: \( 18 \div 2 = 9 \) (3rd factor of 2)
- Now the next prime number is 3. Divide by 3: \( 9 \div 3 = 3 \) (1st factor of 3)
- Divide by 3 again: \( 3 \div 3 = 1 \) (2nd factor of 3)
You now have three 2's and two 3's, confirming that the prime factorization is \( 2^3 \times 3^2 \).
4. **Count and Check:**
Count the total number of factors generated in the factorization. If all calculations have been done correctly, you should have matched the total to the original number upon multiplication.
If both methods confirm that you return to the original number of 72, you can confidently say that your prime factorization is correct.