prime factorization of the numbers 72, 108, 114

The prime factorization of the given numbers can be found by breaking them down into their prime factors.

1. **72**:
   - Start by dividing by the smallest prime number (2):
   \[
   72 \div 2 = 36
   \]
   \[
   36 \div 2 = 18
   \]
   \[
   18 \div 2 = 9
   \]
   Now, divide by the next smallest prime number (3):
   \[
   9 \div 3 = 3
   \]
   \[
   3 \div 3 = 1
   \]
   So, the prime factorization of \(72\) is:
   \[
   72 = 2^3 \times 3^2
   \]

2. **108**:
   - Start by dividing by the smallest prime number (2):
   \[
   108 \div 2 = 54
   \]
   \[
   54 \div 2 = 27
   \]
   Now, divide by the next smallest prime number (3):
   \[
   27 \div 3 = 9
   \]
   \[
   9 \div 3 = 3
   \]
   \[
   3 \div 3 = 1
   \]
   So, the prime factorization of \(108\) is:
   \[
   108 = 2^2 \times 3^3
   \]

3. **114**:
   - Start by dividing by the smallest prime number (2):
   \[
   114 \div 2 = 57
   \]
   Now, divide by the next smallest prime number (3):
   \[
   57 \div 3 = 19
   \]
   \(19\) is a prime number, so we stop here.
   So, the prime factorization of \(114\) is:
   \[
   114 = 2 \times 3 \times 19
   \]

### Summary:
- **72** = \(2^3 \times 3^2\)
- **108** = \(2^2 \times 3^3\)
- **114** = \(2 \times 3 \times 19\)

Would you like more details on any of these factorizations?

### Questions to Explore Further:
1. What are the prime factors of \(360\)?
2. How can prime factorization be used to find the greatest common divisor (GCD) of two numbers?
3. What is the least common multiple (LCM) of \(72\) and \(108\) based on their prime factorizations?
4. How does the Fundamental Theorem of Arithmetic relate to prime factorization?
5. What are the applications of prime factorization in cryptography?

### Tip:
Prime factorization is very useful for finding both the GCD and LCM of numbers quickly.

Explore the prime factorization of the numbers 72, 108, and 114 with a detailed step-by-step solution. Understand how to break down numbers into their prime factors using the prime factorization formula and learn about the Fundamental Theorem of Arithmetic. Suitable for students in Grades 5-7.

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