LCM of `N_(1) and N_(2)` is 100, then find the possible pairs of `N_(1)` & `N_(2)` ?

Options

A. 26
B. 10
C. 18
D. 14

Answer

To find the possible pairs of \( N_1 \) and \( N_2 \) such that their least common multiple (LCM) is 100, we can follow these steps:

### Step 1: Understand the definition of LCM
The least common multiple of two numbers is the smallest number that is a multiple of both numbers.

### Step 2: Factorize the LCM
First, we need to factorize 100 into its prime factors:
\[ 100 = 2^2 \times 5^2 \]

### Step 3: Identify the possible pairs
To find pairs \( (N_1, N_2) \) such that \( \text{LCM}(N_1, N_2) = 100 \), we can use the prime factorization. Each number \( N_1 \) and \( N_2 \) can be expressed in terms of the prime factors of 100.

Let:
\[ N_1 = 2^a \times 5^b \]
\[ N_2 = 2^c \times 5^d \]
where \( a, b, c, d \) are non-negative integers.

### Step 4: Apply the LCM condition
The LCM of \( N_1 \) and \( N_2 \) can be expressed as:
\[ \text{LCM}(N_1, N_2) = 2^{\max(a, c)} \times 5^{\max(b, d)} \]
For this to equal 100, we need:
\[ \max(a, c) = 2 \quad \text{and} \quad \max(b, d) = 2 \]

### Step 5: Determine the values of \( a, b, c, d \)
Now we can find the possible combinations of \( (a, c) \) and \( (b, d) \):

1. For \( \max(a, c) = 2 \):
- Possible pairs \( (a, c) \):
- (2, 0)
- (2, 1)
- (2, 2)
- (1, 2)
- (0, 2)

2. For \( \max(b, d) = 2 \):
- Possible pairs \( (b, d) \):
- (2, 0)
- (2, 1)
- (2, 2)
- (1, 2)
- (0, 2)

### Step 6: Combine the pairs
Now we can combine the pairs of \( (a, c) \) with \( (b, d) \):

- For \( (2, 0) \):
- \( (2, 0), (2, 0) \) → \( (4, 1) \)
- \( (2, 0), (2, 1) \) → \( (4, 5) \)
- \( (2, 0), (2, 2) \) → \( (4, 25) \)
- \( (2, 0), (1, 2) \) → \( (4, 10) \)
- \( (2, 0), (0, 2) \) → \( (4, 100) \)

- For \( (2, 1) \):
- \( (2, 1), (2, 0) \) → \( (4, 5) \)
- \( (2, 1), (2, 1) \) → \( (4, 25) \)
- \( (2, 1), (2, 2) \) → \( (4, 100) \)
- \( (2, 1), (1, 2) \) → \( (4, 10) \)
- \( (2, 1), (0, 2) \) → \( (4, 100) \)

- Continuing this process for all combinations, we will find all pairs that satisfy the LCM condition.

### Final List of Pairs
After calculating all combinations, the possible pairs \( (N_1, N_2) \) such that \( \text{LCM}(N_1, N_2) = 100 \) are:
1. (1, 100)
2. (2, 50)
3. (4, 25)
4. (5, 20)
5. (10, 10)
6. (20, 5)
7. (25, 4)
8. (50, 2)
9. (100, 1)

### Conclusion
Thus, the possible pairs \( (N_1, N_2) \) such that their LCM is 100 are:
- (1, 100)
- (2, 50)
- (4, 25)
- (5, 20)
- (10, 10)
- (20, 5)
- (25, 4)
- (50, 2)
- (100, 1)