To find the LCM (Least Common Multiple) of the numbers 64, 84, and 18 using both the prime factorization method and the common division method, we will follow these steps:
### (a) Prime Factorization Method
1. **Prime Factorization of 64:**
- Divide 64 by the smallest prime number, which is 2.
- \( 64 \div 2 = 32 \)
- \( 32 \div 2 = 16 \)
- \( 16 \div 2 = 8 \)
- \( 8 \div 2 = 4 \)
- \( 4 \div 2 = 2 \)
- \( 2 \div 2 = 1 \)
- So, the prime factorization of 64 is \( 2^6 \).
2. **Prime Factorization of 84:**
- Divide 84 by 2.
- \( 84 \div 2 = 42 \)
- \( 42 \div 2 = 21 \)
- Now divide 21 by the next prime number, which is 3.
- \( 21 \div 3 = 7 \)
- \( 7 \div 7 = 1 \)
- So, the prime factorization of 84 is \( 2^2 \times 3^1 \times 7^1 \).
3. **Prime Factorization of 18:**
- Divide 18 by 2.
- \( 18 \div 2 = 9 \)
- Now divide 9 by 3.
- \( 9 \div 3 = 3 \)
- \( 3 \div 3 = 1 \)
- So, the prime factorization of 18 is \( 2^1 \times 3^2 \).
4. **Combine the Prime Factors:**
- From the factorizations:
- 64: \( 2^6 \)
- 84: \( 2^2 \times 3^1 \times 7^1 \)
- 18: \( 2^1 \times 3^2 \)
- Take the highest power of each prime factor:
- For 2: \( 2^6 \)
- For 3: \( 3^2 \)
- For 7: \( 7^1 \)
5. **Calculate the LCM:**
- LCM = \( 2^6 \times 3^2 \times 7^1 \)
- \( = 64 \times 9 \times 7 \)
- \( = 576 \times 7 \)
- \( = 4032 \)
### (b) Common Division Method
1. **Set up the numbers:**
- Write the numbers 64, 84, and 18 in a row.
2. **Divide by the smallest prime number:**
- Divide by 2:
- \( 64 \div 2 = 32 \)
- \( 84 \div 2 = 42 \)
- \( 18 \div 2 = 9 \)
- Write: \( 32, 42, 9 \)
3. **Continue dividing:**
- Divide by 2 again:
- \( 32 \div 2 = 16 \)
- \( 42 \div 2 = 21 \)
- \( 9 \) remains.
- Write: \( 16, 21, 9 \)
- Divide by 2 again:
- \( 16 \div 2 = 8 \)
- \( 21 \) remains.
- \( 9 \) remains.
- Write: \( 8, 21, 9 \)
- Divide by 2 again:
- \( 8 \div 2 = 4 \)
- \( 21 \) remains.
- \( 9 \) remains.
- Write: \( 4, 21, 9 \)
- Divide by 2 again:
- \( 4 \div 2 = 2 \)
- \( 21 \) remains.
- \( 9 \) remains.
- Write: \( 2, 21, 9 \)
- Divide by 2 again:
- \( 2 \div 2 = 1 \)
- \( 21 \) remains.
- \( 9 \) remains.
- Write: \( 1, 21, 9 \)
- Now divide by 3:
- \( 21 \div 3 = 7 \)
- \( 9 \div 3 = 3 \)
- Write: \( 1, 7, 3 \)
- Finally, divide by 7:
- \( 7 \div 7 = 1 \)
- \( 3 \) remains.
- Write: \( 1, 1, 1 \)
4. **Multiply the divisors:**
- LCM = \( 2^6 \times 3^2 \times 7^1 \)
- \( = 4032 \)
### Final Answer:
The LCM of 64, 84, and 18 is **4032**.
