LCM of `( 2^(3) xx 3 xx5) and ( 2^(4) xx 5xx 7) ` is

Options

A. 40
B. 560
C. 1120
D. 1680

Explanation

To find the LCM of the numbers \(2^3 \times 3 \times 5\) and \(2^4 \times 5 \times 7\), we can follow these steps:

### Step 1: Identify the prime factors and their powers
- For the first number \(A = 2^3 \times 3^1 \times 5^1\):
- The prime factors are:
- \(2\) with power \(3\)
- \(3\) with power \(1\)
- \(5\) with power \(1\)

- For the second number \(B = 2^4 \times 5^1 \times 7^1\):
- The prime factors are:
- \(2\) with power \(4\)
- \(5\) with power \(1\)
- \(7\) with power \(1\)

### Step 2: Determine the highest power of each prime factor
- For the prime factor \(2\):
- The highest power is \(4\) (from \(B\)).

- For the prime factor \(3\):
- The highest power is \(1\) (from \(A\)).

- For the prime factor \(5\):
- The highest power is \(1\) (both \(A\) and \(B\) have \(5^1\)).

- For the prime factor \(7\):
- The highest power is \(1\) (from \(B\)).

### Step 3: Write the LCM using the highest powers
Now, we can write the LCM as:
\[ \text{LCM} = 2^4 \times 3^1 \times 5^1 \times 7^1 \]

### Step 4: Calculate the LCM
1. Calculate \(2^4\):
\[ 2^4 = 16 \]

2. Calculate \(3^1\):
\[ 3^1 = 3 \]

3. Calculate \(5^1\):
\[ 5^1 = 5 \]

4. Calculate \(7^1\):
\[ 7^1 = 7 \]

5. Now multiply these results together:
\[ \text{LCM} = 16 \times 3 \times 5 \times 7 \]

6. Calculate \(16 \times 3\):
\[ 16 \times 3 = 48 \]

7. Calculate \(48 \times 5\):
\[ 48 \times 5 = 240 \]

8. Finally, calculate \(240 \times 7\):
\[ 240 \times 7 = 1680 \]

### Final Answer:
The LCM of \(2^3 \times 3 \times 5\) and \(2^4 \times 5 \times 7\) is \(1680\).

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