Weekly Problem: Puzzle on Number Theory for 26-01-2022

image by author

Solution:

Let

image by author

It is given that n = (100q + r) where

image by author

Now, if 11 | (q + r), then 11 | (99q + q + r), i.e., 11 | (100q + r).

Clearly,

image by author

Thus, P is the number of 5-digit multiples of 11.

Finding the number of 5-digit multiples of 11, we get P = 8181.

Therefore, the answer is 81.