Find the value of n if a= `2^3` x 3, b = 2 x 3 x 5, c = `3^n` x 5 and LCM (a, b, c) = `2^3` x `3^2` x 5.

Options

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

Answer

To find the value of \( n \) given the expressions for \( a \), \( b \), and \( c \), and the LCM of these three numbers, we can follow these steps:

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

### Step 2: Identify the prime factorization of \( a \), \( b \), and \( c \)
From the expressions:
- The prime factorization of \( a \) is \( 2^3 \times 3^1 \times 5^0 \)
- The prime factorization of \( b \) is \( 2^1 \times 3^1 \times 5^1 \)
- The prime factorization of \( c \) is \( 2^0 \times 3^n \times 5^1 \)

### Step 3: Determine the LCM of \( a \), \( b \), and \( c \)
The LCM is found by taking the highest power of each prime factor from all three numbers:
- For \( 2 \): The highest power is \( 2^3 \) (from \( a \))
- For \( 3 \): The highest power is \( 3^2 \) (since we have \( 3^1 \) from both \( a \) and \( b \), and \( 3^n \) from \( c \))
- For \( 5 \): The highest power is \( 5^1 \) (from both \( b \) and \( c \))

Thus, the LCM is:
\[ \text{LCM}(a, b, c) = 2^3 \times 3^2 \times 5^1 \]

### Step 4: Set up the equation for the highest power of \( 3 \)
From the LCM, we see that the highest power of \( 3 \) is \( 2 \). Therefore, we can set up the equation:
\[ \max(1, 1, n) = 2 \]
This means that \( n \) must be equal to \( 2 \) because \( 1 \) is less than \( 2 \).

### Step 5: Solve for \( n \)
From the equation \( n = 2 \), we find that:
\[ n = 2 \]

### Final Answer
Thus, the value of \( n \) is \( 2 \).

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