LCM of `N_(1) and N_(2)` is 200, then find the possible pairs of `N_(1)` & `N_(2)` ?

Options

A. 26
B. 12
C. 18
D. 14

Answer

To find the possible pairs of \( N_1 \) and \( N_2 \) such that their Least Common Multiple (LCM) is 200, we can follow these steps:

### Step 1: Factorize 200
First, we need to find the prime factorization of 200.

\[ 200 = 2^3 \times 5^2 \]

### Step 2: Determine the Number of Factors
To find the total number of factors of a number from its prime factorization, we use the formula:

\[ (\text{exponent of first prime} + 1)(\text{exponent of second prime} + 1) \]

For 200, the exponents are 3 (for 2) and 2 (for 5):

\[ (3 + 1)(2 + 1) = 4 \times 3 = 12 \]

So, 200 has 12 factors.

### Step 3: List the Factors
Now, we will list all the factors of 200:

1. 1
2. 2
3. 4
4. 5
5. 8
6. 10
7. 20
8. 25
9. 40
10. 50
11. 100
12. 200

### Step 4: Find Possible Pairs
Next, we need to find pairs \( (N_1, N_2) \) such that \( \text{LCM}(N_1, N_2) = 200 \). For each factor \( N_1 \), we can find \( N_2 \) by using the relationship:

\[ N_1 \times N_2 = \text{LCM}(N_1, N_2) \times \text{GCD}(N_1, N_2) \]

However, since we are only interested in pairs where the LCM is 200, we can simply pair each factor with its corresponding factor that results in 200 when multiplied.

The pairs are:

1. \( (1, 200) \)
2. \( (2, 100) \)
3. \( (4, 50) \)
4. \( (5, 40) \)
5. \( (8, 25) \)
6. \( (10, 20) \)

### Final Step: Conclusion
Thus, the possible pairs \( (N_1, N_2) \) such that their LCM is 200 are:

- \( (1, 200) \)
- \( (2, 100) \)
- \( (4, 50) \)
- \( (5, 40) \)
- \( (8, 25) \)
- \( (10, 20) \)