The sum of two numbers is 216 and their H.C.F is 27. The numbers are

Options

A. 54, 162
B. 108, 108
C. 27, 189
D. None of these

Answer

To solve the problem, we need to find two numbers whose sum is 216 and whose highest common factor (H.C.F) is 27. Let's break down the solution step by step.

### Step 1: Understanding the relationship between the numbers and their H.C.F.
Let the two numbers be represented as:
- First number = 27p
- Second number = 27q
Where p and q are co-prime numbers (i.e., they have no common factors other than 1).

### Step 2: Set up the equation based on the sum of the numbers.
According to the problem, the sum of the two numbers is given as:
\[ 27p + 27q = 216 \]

### Step 3: Simplify the equation.
We can factor out 27 from the left side:
\[ 27(p + q) = 216 \]
Now, divide both sides by 27:
\[ p + q = \frac{216}{27} \]
Calculating the right side:
\[ p + q = 8 \]

### Step 4: Find co-prime pairs that sum to 8.
Now we need to find pairs of co-prime numbers (p, q) that add up to 8. The possible pairs are:
- (1, 7)
- (3, 5)

### Step 5: Calculate the actual numbers using the pairs.
For the pair (1, 7):
- First number = 27 * 1 = 27
- Second number = 27 * 7 = 189

For the pair (3, 5):
- First number = 27 * 3 = 81
- Second number = 27 * 5 = 135

### Step 6: Verify the sums.
Now, we check the sums:
- For (1, 7): 27 + 189 = 216
- For (3, 5): 81 + 135 = 216

Both pairs satisfy the sum condition. However, since we are looking for the numbers that correspond to the H.C.F. being 27, we can conclude that the first pair (27, 189) is the correct answer.

### Final Answer:
The two numbers are **27 and 189**.

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