To find the LCM of the numbers 45, 78, and 15 using both the prime factorization method and the common division method, we will follow these steps:
### (a) Prime Factorization Method
**Step 1: Prime Factorization of Each Number**
- **For 45**:
- Divide by 3: \( 45 \div 3 = 15 \)
- Divide by 3 again: \( 15 \div 3 = 5 \)
- Finally, divide by 5: \( 5 \div 5 = 1 \)
- So, the prime factorization of 45 is \( 3^2 \times 5^1 \).
- **For 78**:
- Divide by 2: \( 78 \div 2 = 39 \)
- Divide by 3: \( 39 \div 3 = 13 \)
- Finally, divide by 13: \( 13 \div 13 = 1 \)
- So, the prime factorization of 78 is \( 2^1 \times 3^1 \times 13^1 \).
- **For 15**:
- Divide by 3: \( 15 \div 3 = 5 \)
- Finally, divide by 5: \( 5 \div 5 = 1 \)
- So, the prime factorization of 15 is \( 3^1 \times 5^1 \).
**Step 2: List the Prime Factors**
- From the factorizations, we have:
- \( 45 = 3^2 \times 5^1 \)
- \( 78 = 2^1 \times 3^1 \times 13^1 \)
- \( 15 = 3^1 \times 5^1 \)
**Step 3: Identify the Highest Powers of Each Prime Factor**
- The prime factors are 2, 3, 5, and 13.
- The highest powers are:
- \( 2^1 \) from 78
- \( 3^2 \) from 45
- \( 5^1 \) from both 45 and 15
- \( 13^1 \) from 78
**Step 4: Calculate the LCM**
- LCM = \( 2^1 \times 3^2 \times 5^1 \times 13^1 \)
- LCM = \( 2 \times 9 \times 5 \times 13 \)
- LCM = \( 2 \times 9 = 18 \)
- \( 18 \times 5 = 90 \)
- \( 90 \times 13 = 1170 \)
### (b) Common Division Method
**Step 1: Set Up the Division Table**
- Write the numbers: 45, 78, and 15.
**Step 2: Divide by the Smallest Prime Number**
- Start with 2 (not applicable since 45 is odd).
- Next, divide by 3:
- \( 45 \div 3 = 15 \)
- \( 78 \div 3 = 26 \)
- \( 15 \div 3 = 5 \)
**Step 3: Continue Dividing**
- Now, we have: 15, 26, 5.
- Divide by 5:
- \( 15 \div 5 = 3 \)
- \( 26 \div 5 \) (not divisible, keep 26)
- \( 5 \div 5 = 1 \)
**Step 4: Continue Dividing**
- Now we have: 3, 26, 1.
- Divide by 13:
- \( 3 \div 1 \) (keep 3)
- \( 26 \div 13 = 2 \)
- \( 1 \div 1 = 1 \)
**Step 5: Final Division**
- Now we have: 1, 2, 1.
- Divide by 2:
- \( 1 \div 1 \) (keep 1)
- \( 2 \div 2 = 1 \)
- \( 1 \div 1 = 1 \)
**Step 6: Calculate the LCM**
- LCM = \( 3^2 \times 5^1 \times 2^1 \times 13^1 \)
- LCM = \( 2 \times 3 \times 3 \times 5 \times 13 = 1170 \)
### Final Answer
The LCM of 45, 78, and 15 is **1170**.
