The sum of two 2- diigit numbers is 132 . If their HCF is 11, the numbers are

Options

A. 55,77
B. 44,88
C. 33,99
D. 22,110

Answer

To solve the problem step by step, we will follow the reasoning provided in the video transcript.

### Step 1: Understanding the problem
We know that the sum of two two-digit numbers is 132, and their highest common factor (HCF) is 11.

### Step 2: Express the numbers in terms of their HCF
Since the HCF of the two numbers is 11, we can express the two numbers as:
- First number = 11x
- Second number = 11y

### Step 3: Set up the equation based on the sum
According to the problem, the sum of the two numbers is 132. Therefore, we can write the equation:
\[ 11x + 11y = 132 \]

### Step 4: Simplify the equation
We can factor out 11 from the left side:
\[ 11(x + y) = 132 \]
Now, divide both sides by 11:
\[ x + y = \frac{132}{11} \]
\[ x + y = 12 \]

### Step 5: Find possible pairs (x, y)
Now we need to find pairs of integers (x, y) such that their sum is 12. The possible pairs are:
- (1, 11)
- (2, 10)
- (3, 9)
- (4, 8)
- (5, 7)
- (6, 6)

### Step 6: Check for HCF
We need to check which of these pairs gives an HCF of 11 when multiplied by 11:
1. For (1, 11): HCF(11, 121) = 11
2. For (2, 10): HCF(22, 110) = 11
3. For (3, 9): HCF(33, 99) = 33
4. For (4, 8): HCF(44, 88) = 44
5. For (5, 7): HCF(55, 77) = 11
6. For (6, 6): HCF(66, 66) = 66

### Step 7: Identify valid pairs
From the pairs checked, the valid pairs that give an HCF of 11 are:
- (1, 11) → Numbers: 11 and 121 (not valid, since 121 is not a two-digit number)
- (2, 10) → Numbers: 22 and 110 (not valid, since 110 is not a two-digit number)
- (5, 7) → Numbers: 55 and 77 (valid, both are two-digit numbers)

### Step 8: Conclusion
The two-digit numbers that satisfy the conditions of the problem are:
- 55 and 77