Find the LCM of the following numbers using: (a) prime factorisation (b) common division
64, 84 and 180

Answer

To find the LCM of the numbers 64, 84, and 180 using two methods: prime factorization and common division, we will follow the steps outlined below.

### (a) Prime Factorization Method

**Step 1: Prime Factorization of Each Number**
1. **Factorizing 64:**
- 64 = 2 × 32
- 32 = 2 × 16
- 16 = 2 × 8
- 8 = 2 × 4
- 4 = 2 × 2
- 2 = 2 × 1
- Therefore, \( 64 = 2^6 \)

2. **Factorizing 84:**
- 84 = 2 × 42
- 42 = 2 × 21
- 21 = 3 × 7
- 7 = 7 × 1
- Therefore, \( 84 = 2^2 × 3^1 × 7^1 \)

3. **Factorizing 180:**
- 180 = 2 × 90
- 90 = 2 × 45
- 45 = 3 × 15
- 15 = 3 × 5
- 5 = 5 × 1
- Therefore, \( 180 = 2^2 × 3^2 × 5^1 \)

**Step 2: Identify the Highest Powers of Each Prime Factor**
- For 2: The highest power is \( 2^6 \) (from 64)
- For 3: The highest power is \( 3^2 \) (from 180)
- For 5: The highest power is \( 5^1 \) (from 180)
- For 7: The highest power is \( 7^1 \) (from 84)

**Step 3: Calculate the LCM**
- LCM = \( 2^6 × 3^2 × 5^1 × 7^1 \)

**Step 4: Perform the Calculation**
- \( LCM = 64 × 9 × 5 × 7 \)
- \( LCM = 64 × 9 = 576 \)
- \( LCM = 576 × 5 = 2880 \)
- \( LCM = 2880 × 7 = 20160 \)

Thus, the LCM of 64, 84, and 180 is **20160**.

### (b) Common Division Method

**Step 1: Set Up the Division Table**
- Write the numbers 64, 84, and 180 in a row.

**Step 2: Divide by the Smallest Prime Number**
- Divide by 2:
- \( 64 ÷ 2 = 32 \)
- \( 84 ÷ 2 = 42 \)
- \( 180 ÷ 2 = 90 \)

**Step 3: Continue Dividing**
- Divide by 2 again:
- \( 32 ÷ 2 = 16 \)
- \( 42 ÷ 2 = 21 \)
- \( 90 ÷ 2 = 45 \)

- Divide by 2 again:
- \( 16 ÷ 2 = 8 \)
- \( 21 \) remains the same.
- \( 45 \) remains the same.

- Divide by 2 again:
- \( 8 ÷ 2 = 4 \)
- \( 21 \) remains the same.
- \( 45 \) remains the same.

- Divide by 2 again:
- \( 4 ÷ 2 = 2 \)
- \( 21 \) remains the same.
- \( 45 \) remains the same.

- Divide by 2 again:
- \( 2 ÷ 2 = 1 \)
- \( 21 \) remains the same.
- \( 45 \) remains the same.

- Now divide by 3:
- \( 21 ÷ 3 = 7 \)
- \( 45 ÷ 3 = 15 \)

- Divide by 3 again:
- \( 15 ÷ 3 = 5 \)

- Now divide by 5:
- \( 5 ÷ 5 = 1 \)

- Finally, divide by 7:
- \( 7 ÷ 7 = 1 \)

**Step 4: Write Down the Prime Factors**
- The prime factors obtained are:
- \( 2^6, 3^2, 5^1, 7^1 \)

**Step 5: Calculate the LCM**
- LCM = \( 2^6 × 3^2 × 5^1 × 7^1 = 20160 \)

Thus, the LCM of 64, 84, and 180 using the common division method is also **20160**.