Let S be the smallest positive multiple of 15 that comprises exactly 3k digits with k ‘0’s, k ‘3’s and k ‘8’s. Find the remainder when S is divided by 8.
Answer:
0
- If a number is a multiple of 15, it is a multiple of 3 and 5 both.
We are given that S is the smallest positive multiple of 15 which comprises exactly 3k digits. Also, S has k ‘0’s, k‘3’s, and k‘8’s.
Observe that S must end with 0 as it is a multiple of 5. - The sum of all the digits of S=k×0+k×3+k×8=3k+8k=11k
Since S is a multiple of 3, the sum of all its digits must be a multiple of 3.
The smallest value of k such that 11k is a multiple of 3 is 3. Therefore, there are 3‘0’s, 3‘3’s, and 3‘8’s in S.
⟹S=300338880 The remainder when S is divided by 8 = Remainder of (Last 3 digits of S÷8)
= Remainder of (880÷8)
=0- Hence, the remainder when S is divided by 8 is 0.