The LCM of (54)8 and (20)8 is

Options

A. (260)8
B. (360)8
C. (54)8
D. (108)8

Answer

To find the LCM of (54)₈ and (20)₈, we will follow these steps:

### Step 1: Convert the octal numbers to decimal.
1. **Convert (54)₈ to decimal:**
- The number (54)₈ can be broken down as:
- \(5 \times 8^1 + 4 \times 8^0\)
- \(= 5 \times 8 + 4 \times 1\)
- \(= 40 + 4 = 44\)

2. **Convert (20)₈ to decimal:**
- The number (20)₈ can be broken down as:
- \(2 \times 8^1 + 0 \times 8^0\)
- \(= 2 \times 8 + 0 \times 1\)
- \(= 16 + 0 = 16\)

### Step 2: Find the LCM of the decimal numbers 44 and 16.
1. **Factor the numbers:**
- The prime factorization of 44 is:
- \(44 = 2^2 \times 11\)
- The prime factorization of 16 is:
- \(16 = 2^4\)

2. **Determine the LCM:**
- The LCM is found by taking the highest power of each prime factor:
- For the prime factor 2: max power is \(2^4\)
- For the prime factor 11: max power is \(11^1\)
- Therefore, the LCM is:
- \(LCM = 2^4 \times 11^1 = 16 \times 11 = 176\)

### Step 3: Convert the LCM back to octal.
1. **Convert 176 to octal:**
- Divide 176 by 8:
- \(176 \div 8 = 22\) with a remainder of \(0\)
- Divide 22 by 8:
- \(22 \div 8 = 2\) with a remainder of \(6\)
- Divide 2 by 8:
- \(2 \div 8 = 0\) with a remainder of \(2\)
- Now, read the remainders from bottom to top: \(260\)

### Final Answer:
The LCM of (54)₈ and (20)₈ is (260)₈.

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