Question

The sum of the two-digit number is \[132\]. If their HCF is \[11\], the numbers are:
A. \[55\], \[77\]
B. \[44\], \[88\]
C. \[33\], \[99\]
D. \[22\], \[110\]

Answer 1

Hint: Here we will use the rule of HCF of two numbers which states that if HCF of two numbers is given then the numbers can be written in the form of their product with HCF as shown below:
Suppose the HCF of any two numbers \[X\] and \[Y\] is \[k\], then the numbers can written as \[ka\] and \[kb\] where \[a\] and \[b\] are co-prime to each other.

Complete step-by-step solution:
Step 1: Let the numbers are \[X\] and \[Y\] so as per the information given in the question, we can write as below:
\[X + Y = 132\]
Also, we know that the HCF of \[X\] and \[Y\] is \[11\] , meaning the numbers are divisible by \[11\].
By using the property of HCF, we can write the two numbers as below:
\[ \Rightarrow X = 11a\] and
\[Y = 11b\] , where \[a\] and
\[b\] are co-prime to each other
Step 2: By substituting the values of \[X = 11a\] and
\[Y = 11b\] into the equation \[X + Y = 132\], we get:
\[ \Rightarrow 11a + 11b = 132\]
By taking \[11\] common into the LHS side of the above equation, we get:
\[ \Rightarrow 11\left( {a + b} \right) = 132\]
By bringing \[11\] into the RHS side of the equation and dividing it with
\[132\], we get:
\[ \Rightarrow a + b = 12\]
Step 3: Now, we will write the possible pairs of the two numbers whose sum equals to \[12\], as shown below:
Pairs are:
\[\left( {1,11} \right)\], \[\left( {2,10} \right)\], \[\left( {3,9} \right)\],\[\left( {4,8} \right)\], \[\left( {5,7} \right)\], \[\left( {6,6} \right)\]
Now, from these all pairs only two pairs are there which are prime to each other,
\[\left( {1,11} \right)\] and \[\left( {5,7} \right)\].
Now by using these two pairs we will check which pair is the correct one. For calculating the numbers, we will substitute these values in the expression
\[X = 11a\] and \[Y = 11b\].
For the pair \[\left( {1,11} \right)\], the numbers are as below:
\[X = 11 \times 1\] and \[Y = 11 \times 11\]. By simplifying the expressions, we get:
\[ \Rightarrow X = 11\] and \[Y = 121\]
But the numbers should be of two digits as asked in the question. So, this pair is not possible.
For the pair \[\left( {5,7} \right)\], the numbers are as below:
\[X = 5 \times 11\] and \[Y = 7 \times 11\]. By simplifying the expressions, we get:
\[ \Rightarrow X = 55\] and \[Y = 77\]

Option A is correct.

Note: Students need to take while forming the pairs of the numbers, also sometimes students try to guess the answer from the given options. But in this question, all the options given have a sum of \[132\]. So, you should follow the full procedure as explained in the answer. If only one option has the of \[132\] then we can directly select the option with the sum of \[132\] as our answer.