LCM of `N_(1) and N_(2)` is 300, then find the possible pairs of `N_(1)` & `N_(2)` ?

Options

A. 38
B. 39
C. 40
D. None of these

Answer

To find the possible pairs of \( N_1 \) and \( N_2 \) such that their Least Common Multiple (LCM) is 300, we can follow these steps:

### Step 1: Factorize 300
First, we need to find the prime factorization of 300.

\[ 300 = 3 \times 100 = 3 \times 10^2 = 3 \times (2 \times 5)^2 = 3^1 \times 2^2 \times 5^2 \]

So, the prime factorization of 300 is:

\[ 300 = 2^2 \times 3^1 \times 5^2 \]

### Step 2: Identify Divisors of 300
Next, we will find all the divisors of 300. The divisors can be formed by taking all combinations of the prime factors:

- For \( 2^0, 2^1, 2^2 \) (which gives us 1, 2, 4)
- For \( 3^0, 3^1 \) (which gives us 1, 3)
- For \( 5^0, 5^1, 5^2 \) (which gives us 1, 5, 25)

Now, we can list the divisors by multiplying these combinations:

- \( 1 \)
- \( 2 \)
- \( 3 \)
- \( 4 \)
- \( 5 \)
- \( 6 \)
- \( 10 \)
- \( 12 \)
- \( 15 \)
- \( 20 \)
- \( 25 \)
- \( 30 \)
- \( 50 \)
- \( 60 \)
- \( 75 \)
- \( 100 \)
- \( 150 \)
- \( 300 \)

### Step 3: Find Pairs of Divisors
Now we need to find pairs of these divisors \( (N_1, N_2) \) such that:

\[ \text{LCM}(N_1, N_2) = 300 \]

We will check pairs systematically:

1. \( (1, 300) \)
2. \( (2, 150) \)
3. \( (3, 100) \)
4. \( (4, 75) \)
5. \( (5, 60) \)
6. \( (6, 50) \)
7. \( (10, 30) \)
8. \( (12, 25) \)
9. \( (15, 20) \)

### Step 4: Verify LCM for Each Pair
We need to verify that the LCM of each pair equals 300:

- **For \( (1, 300) \)**: LCM = 300
- **For \( (2, 150) \)**: LCM = 300
- **For \( (3, 100) \)**: LCM = 300
- **For \( (4, 75) \)**: LCM = 300
- **For \( (5, 60) \)**: LCM = 300
- **For \( (6, 50) \)**: LCM = 300
- **For \( (10, 30) \)**: LCM = 300
- **For \( (12, 25) \)**: LCM = 300
- **For \( (15, 20) \)**: LCM = 300

### Conclusion
The possible pairs \( (N_1, N_2) \) such that their LCM is 300 are:

1. \( (1, 300) \)
2. \( (2, 150) \)
3. \( (3, 100) \)
4. \( (4, 75) \)
5. \( (5, 60) \)
6. \( (6, 50) \)
7. \( (10, 30) \)
8. \( (12, 25) \)
9. \( (15, 20) \)