What will be the smallest value of m such that the five - digit number 1m813 is divisible by 11?

Correct option is C

Given:

Five-digit number: 1m813

Need: Smallest value of digit (m) such that the number is divisible by 11.

Concept Used:

Divisibility rule of 11: (Sum of digits at odd places) - (Sum of digits at even places) must be 0 or a multiple of 11.

Solution:

Positions from right:

Odd places: 3 (1st), 8 (3rd), 1 (5th)

S(odd) = 3 + 8 + 1 = 12

Even places: 1 (2nd), (m) (4th)

S(even) = 1 + m

Difference: |S(odd) - S(even)| = |12 - (1 + m)| = |11 - m|

For divisibility by 11:

|11 - m| = 0, 11, 22, ......

Case 1: 11 - m = 0 $$\Rightarrow$$​ m = 11) (Not valid, since (m) must be a digit 0 – 9)

Case 2: 11 - m = 11 $$\Rightarrow$$​ m = 0 (Valid)

Case 3: 11 - m = -11 $$\Rightarrow$$​ m = 22 (Not valid)

Thus the smallest valid digit is 0