Let be the smallest positive multiple of that comprises exactly digits with k ‘0’s, k ‘3’s and k ‘8’s. Find the remainder when is divided by .
Answer:
- If a number is a multiple of , it is a multiple of and both.
We are given that is the smallest positive multiple of which comprises exactly digits. Also, has ‘’, ‘’ and ‘’
Observe that must end with as it is a multiple of . - The sum of all the digits of
Since is a multiple of , the sum of all its digits must be a multiple of .
The smallest value of such that is a multiple of is . Therefore, there are ‘’, ‘’ and ‘’ in .
The remainder when is divided by Remainder of (Last digits of )
Remainder of ()
- Hence, the remainder when is divided by is .