If the Least Common Multiple of 56, 57 and 58 is K, then what will be the Least Common Multiple of 56, 57, 58 and 59?

Options

A. 177 K
B. 59 K
C. 57 K
D. 56 K

Answer

To find the Least Common Multiple (LCM) of the numbers 56, 57, 58, and 59, we can follow these steps:

### Step 1: Understand the LCM
The LCM of a set of numbers is the smallest number that is a multiple of each of the numbers in the set.

### Step 2: Prime Factorization
First, we need to find the prime factorization of each number:
- **56**: \( 56 = 2^3 \times 7^1 \)
- **57**: \( 57 = 3^1 \times 19^1 \)
- **58**: \( 58 = 2^1 \times 29^1 \)
- **59**: \( 59 = 59^1 \) (since 59 is a prime number)

### Step 3: Identify the Highest Powers of Each Prime
Next, we identify the highest powers of each prime number present in the factorizations:
- For \( 2 \): The highest power is \( 2^3 \) (from 56).
- For \( 3 \): The highest power is \( 3^1 \) (from 57).
- For \( 7 \): The highest power is \( 7^1 \) (from 56).
- For \( 19 \): The highest power is \( 19^1 \) (from 57).
- For \( 29 \): The highest power is \( 29^1 \) (from 58).
- For \( 59 \): The highest power is \( 59^1 \) (from 59).

### Step 4: Calculate the LCM
Now we can calculate the LCM by multiplying the highest powers of all prime factors:
\[ \text{LCM} = 2^3 \times 3^1 \times 7^1 \times 19^1 \times 29^1 \times 59^1 \]

### Step 5: Find the LCM of 56, 57, 58, and 59
Since 59 is a prime number and does not share any prime factors with 56, 57, or 58, the LCM of 56, 57, and 58 (which is K) will remain the same when we include 59. Therefore, the LCM of 56, 57, 58, and 59 is:
\[ \text{LCM}(56, 57, 58, 59) = \text{LCM}(K, 59) = K \times 59 \]

### Conclusion
Thus, the Least Common Multiple of 56, 57, 58, and 59 is \( K \times 59 \).