Question

A 3-digit number 4a3 is added to another 3-digit number 984 to give the four-digit number 13b7 which is divisible by 11 Then (a + b) is
A.10
B.11
C.12
D.13

Answer 1

Hint: Here in this question, we have to find the value of the sum of unknowns i.e., \[\left( {a + b} \right)\] . To solve this first by the given condition and by comparing we have to write the relations between the unknowns a and b and later by the given hint 13b7 which is divisible by 11 using this and by further simplification, we get the required solution

Complete step by step solution:
Consider the given data in question:
A 3-digit number 4a3 is added to another 3-digit number 984 to give the four-digit number 13b7
This is can be written in the mathematical form as:
 \[ \Rightarrow 4a3 + 984 = 13b7\]
Now, by comparing the middle term of the above expression, we can write the relation between the unknowns a and b, i.e.,
 \[ \Rightarrow a + 8 = b\]
Subtract both side by 8, then
 \[ \Rightarrow a + 8 - 8 = b - 8\]
On simplification, we get
 \[ \Rightarrow a = b - 8\] ----------(1)
Consider a given hint i.e., \[13b7\] is divisible by 11. Then by rule of divisibility by 11
 \[ \Rightarrow \left( {1 + b} \right) - \left( {3 + 7} \right)\] is \[0\] or divisible by 11
 \[ \Rightarrow \left( {1 + b} \right) - \left( {3 + 7} \right)\]
 \[ \Rightarrow 1 + b - 3 - 7\]
On simplification, we get
 \[ \Rightarrow 1 + b - 10\]
 \[ \Rightarrow \left( {b - 9} \right)\] is \[0\] or divisible by 11-------(2)
Now, assume that
 \[ \Rightarrow \left( {b - 9} \right) = 0\]
 \[ \Rightarrow b - 9 = 0\]
Add 9 on both side, then
 \[ \Rightarrow b = 9\]
Substitute the value of b in equation (1), we have
 \[ \Rightarrow a = 9 - 8\]
On simplification, we get
 \[ \Rightarrow a = 1\]
If \[a = 1\] and \[b = 9\] , then the value of
 \[ \Rightarrow \,\,\left( {a + b} \right) = \left( {1 + 9} \right) = 10\]
10 is there in the given choices.
Hence option A is the correct answer.
So, the correct answer is “Option A”.

Note: To find the number which is divisible by 11 or not by using a rule of divisibility by 11. A number passes the test for 11 if the difference of the sums of alternating digits is divisible by 11 and remember that any number evenly divides 0. And the main thing on this type of question is comparison of terms which locate the place and observe it seriously.