Find the value of 'n' if

`a=2^(3)xx3`
`b=2xx3xx4`
`c=3^(n)xx5`

LCM (a, b, c) = `2^(3)xx3^(2)xx5`.

Options

A. 1
B. 2
C. 3
D. 4

Answer

To find the value of 'n' given the equations for a, b, and c, and the LCM of these values, we can follow these steps:

### Step 1: Write down the values of a, b, and c.
Given:
- \( a = 2^3 \times 3 \)
- \( b = 2 \times 3 \times 4 \)
- \( c = 3^n \times 5 \)

### Step 2: Simplify the expression for b.
We can express 4 as \( 2^2 \):
- \( b = 2 \times 3 \times 4 = 2 \times 3 \times (2^2) = 2^3 \times 3 \)

### Step 3: Write down the LCM expression.
The LCM of a, b, and c is given as:
- \( \text{LCM}(a, b, c) = 2^3 \times 3^2 \times 5 \)

### Step 4: Identify the maximum powers of each prime factor.
To find the LCM, we take the highest power of each prime factor from a, b, and c:
- For \( 2 \): The maximum power is \( 2^3 \) (from both a and b).
- For \( 3 \): The maximum power is \( \max(1, 1, n) \) (from a, b, and c).
- For \( 5 \): The maximum power is \( 1 \) (only from c).

### Step 5: Set up the equation based on the LCM.
From the LCM expression, we can equate the powers:
- For \( 2 \): \( 2^3 \) matches \( 2^3 \) (no new information).
- For \( 3 \): \( \max(1, 1, n) = 2 \)
- For \( 5 \): \( 0 = 1 \) (no contribution from a and b).

### Step 6: Solve for n.
From the equation \( \max(1, 1, n) = 2 \):
- This implies that \( n \) must be equal to \( 2 \) since \( 1 < 2 \).

### Conclusion:
Thus, the value of \( n \) is:
- \( n = 2 \)