Question

Which of the following numbers is divisible by$11$.

     A. \[74,396\]

     B. $45,403$

     C. $60,390$

     D. $98,498$

Answer 1

Hint- To answer this problem we will use the definition as if after subtraction of the sum of the digit at even places from the sum of the digit at odd places the number is divisible by $11$ we get $0$or $11$. So we will use this definition to test all the options.

Complete step-by-step answer:
As we all know
Divisibility law of $11$: The sum of the digits at even places is subtracted at odd places from the number of odd digits. Either the result is $0$ or divisible by $11$.
We will now test through alternative
By taking option A.
A ) \[74,396\]
Here digits at odd places are \[7,3,6\]
and digits at even places are \[4,9\]
By using the divisibility rule of \[11\]
Sum of digits at odd places \[ - \] Sum of digits at even places \[ = 0\]or \[11\]
Substitute the values in above formula , we have
\[\left( {7 + 3 + 6} \right){\text{ }} - \left( {4 + 9} \right){\text{ }} = {\text{ }}16 - 13{\text{ }} = {\text{ }}3\]
Here the result comes out as \[3\] therefore \[74,396\] is not divisible by \[11\].

Now take option B
B) $45,403$
Here digits at odd places are \[4,4,3\]
and digits at even places are \[5,0\]
By using the divisibility rule of \[11\]
Sum of digits at odd places \[ - \] Sum of digits at even places \[ = 0\]or \[11\]
Substitute the values in above formula , we have
\[\begin{array}{*{20}{l}}
  \; \\
  {\left( {4 + 4 + 3} \right) - \left( {5 + 0} \right) = 11 - 5 = 6}
\end{array}\]

Here the result comes out as \[6\] therefore $45,403$ is not divisible by \[11\].

By taking option C
C) $60,390$
Here digits at odd places are \[6,3,0\]
and digits at even places are \[0,9\]
By using the divisibility rule of \[11\]
Sum of digits at odd places \[ - \] Sum of digits at even places \[ = 0\]or \[11\]
Substitute the values in above formula , we have
\[60,390 = \left( {6 + 3} \right) - 9 = 9 - 9 = 0\]
Here the result comes out as \[0\] therefore $60,390$ is divisible by \[11\] .

Considering option D.
D) $98,498$
Here digits at odd places are \[9,4,8\]
and digits at even places are \[8,9\]
By using the divisibility rule of \[11\]
Sum of digits at odd places \[ - \] Sum of digits at even places \[ = 0\]or \[11\]
Substitute the values in above formula , we have
\[98,498 = \left( {9 + 4 + 8} \right) - \left( {8 + 9} \right) = 21 - 17 = 4\]
Here the result comes out as$4$therefore $98,498$ is not divisible by \[11\].
Hence the correct answer is option C.

Note- From the alternating digit number. The result has to be divisible by \[11\]. Add the digits from right to left into blocks of two. The result has to be divisible by \[11\]. Subtract the remaining last digit. The result has to be divisible by \[11\].