Find the two numbers whose LCM is 78 and HCF is 13?

Options

A. 58 and 13 or 16 and 29
B. 68 and 23 or 36 and 49
C. 18 and 73 or 56 and 93
D. 78 and 13 or 26 and 39

Answer

To find the two numbers whose LCM is 78 and HCF is 13, we can follow these steps:

### Step 1: Use the relationship between LCM, HCF, and the two numbers
We know that the product of the two numbers (let's call them \( a \) and \( b \)) is equal to the product of their LCM and HCF. This can be expressed as:
\[ a \times b = \text{LCM}(a, b) \times \text{HCF}(a, b) \]
Substituting the given values:
\[ a \times b = 78 \times 13 \]

### Step 2: Calculate the product
Now, let's calculate \( 78 \times 13 \):
\[ 78 \times 13 = 1014 \]
So, we have:
\[ a \times b = 1014 \]

### Step 3: Prime factorization of the product
Next, we will perform the prime factorization of 1014 to help us find the two numbers:
\[ 1014 = 2 \times 3 \times 13 \times 13 \]
This can also be written as:
\[ 1014 = 2 \times 3 \times 13^2 \]

### Step 4: Identify the common factor
Since we know that the HCF of \( a \) and \( b \) is 13, we can express \( a \) and \( b \) in terms of their common factor:
Let:
\[ a = 13m \quad \text{and} \quad b = 13n \]
where \( m \) and \( n \) are co-prime (i.e., HCF(m, n) = 1).

### Step 5: Substitute and simplify
Now substituting \( a \) and \( b \) into the product:
\[ (13m) \times (13n) = 1014 \]
This simplifies to:
\[ 169mn = 1014 \]
Dividing both sides by 169 gives:
\[ mn = \frac{1014}{169} = 6 \]

### Step 6: Find pairs of \( m \) and \( n \)
Now we need to find pairs of \( m \) and \( n \) such that their product is 6. The pairs of factors of 6 are:
1. \( (1, 6) \)
2. \( (2, 3) \)

### Step 7: Calculate \( a \) and \( b \)
Now we can find the corresponding values of \( a \) and \( b \):

1. For \( (m, n) = (1, 6) \):
\[ a = 13 \times 1 = 13, \quad b = 13 \times 6 = 78 \]

2. For \( (m, n) = (2, 3) \):
\[ a = 13 \times 2 = 26, \quad b = 13 \times 3 = 39 \]

### Final Result
Thus, the two pairs of numbers whose LCM is 78 and HCF is 13 are:
1. \( (13, 78) \)
2. \( (26, 39) \)