The sum of two numbers is 216 and their HCF is 27 . How many pairs of such numbers are there ?

Options

A. 1
B. 2
C. 3
D. 0

Answer

To solve the problem step by step, we need to find how many pairs of numbers exist such that their sum is 216 and their highest common factor (HCF) is 27.

### Step 1: Understand the relationship between the numbers
Given:
- The sum of two numbers, \( a + b = 216 \)
- The HCF of the two numbers, \( \text{HCF}(a, b) = 27 \)

### Step 2: Express the numbers in terms of their HCF
Since the HCF of the two numbers is 27, we can express the two numbers as:
- \( a = 27x \)
- \( b = 27y \)

where \( x \) and \( y \) are coprime (i.e., their HCF is 1).

### Step 3: Substitute into the sum equation
Substituting \( a \) and \( b \) into the sum equation:
\[ 27x + 27y = 216 \]

### Step 4: Simplify the equation
Factoring out 27 from the left side:
\[ 27(x + y) = 216 \]
Now, divide both sides by 27:
\[ x + y = \frac{216}{27} = 8 \]

### Step 5: Find pairs of coprime integers
Now we need to find pairs of integers \( (x, y) \) such that:
- \( x + y = 8 \)
- \( \text{HCF}(x, y) = 1 \)

The possible pairs of integers that sum to 8 are:
1. \( (1, 7) \)
2. \( (2, 6) \)
3. \( (3, 5) \)
4. \( (4, 4) \)

### Step 6: Check which pairs are coprime
Now we check which of these pairs are coprime:
- \( (1, 7) \): HCF is 1 (Coprime)
- \( (2, 6) \): HCF is 2 (Not coprime)
- \( (3, 5) \): HCF is 1 (Coprime)
- \( (4, 4) \): HCF is 4 (Not coprime)

### Step 7: Count the valid pairs
The valid coprime pairs are:
1. \( (1, 7) \)
2. \( (3, 5) \)

Thus, there are **2 pairs** of such numbers.

### Final Answer
The number of pairs of such numbers is **2**.

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