Find the number of pairs of natural numbers with LCM as 56

Options

A. 3
B. 4
C. 10
D. Can't be determined

Answer

To find the number of pairs of natural numbers (A, B) such that their least common multiple (LCM) is 56, we can follow these steps:

### Step 1: Identify the divisors of 56
First, we need to find all the divisors of 56. The divisors are the numbers that can divide 56 without leaving a remainder.

**Divisors of 56**:
1, 2, 4, 7, 8, 14, 28, 56

### Step 2: List all possible pairs of divisors
Next, we will list all possible pairs (A, B) of these divisors. Since the order of the pair does not matter (i.e., (A, B) is the same as (B, A)), we will consider each pair only once.

### Step 3: Calculate the LCM for each pair
Now, we will calculate the LCM for each pair to check if it equals 56.

1. (1, 56) → LCM = 56
2. (2, 28) → LCM = 56
3. (4, 14) → LCM = 56
4. (7, 8) → LCM = 56
5. (2, 56) → LCM = 56
6. (4, 56) → LCM = 56
7. (7, 56) → LCM = 56
8. (8, 56) → LCM = 56
9. (14, 56) → LCM = 56
10. (28, 56) → LCM = 56

### Step 4: Count the valid pairs
From the calculations above, we can see that the valid pairs where the LCM is 56 are:
- (1, 56)
- (2, 28)
- (4, 14)
- (7, 8)
- (2, 56)
- (4, 56)
- (7, 56)
- (8, 56)
- (14, 56)
- (28, 56)

### Conclusion
After listing all pairs and checking their LCM, we find that there are a total of **10 pairs** of natural numbers (A, B) such that LCM(A, B) = 56.

### Final Answer
The number of pairs of natural numbers with LCM as 56 is **10**.

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