Let S be the smallest positive multiple of 15 that comprises exactly 3k digits with k โ0โs, k โ3โs and k โ8โs. 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 โ0โs, kโ3โs, and kโ8โs.
Observe that S must end with 0 as it is a multiple of 5. - The sum of all the digits of S=kร0+kร3+kร8=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 3โ0โs, 3โ3โs, and 3โ8โs in S.
โนS=300338880 The remainder when S is divided by 8 = Remainder of (Last 3 digits of Sรท8)
= Remainder of (880รท8)
=0- Hence, the remainder when S is divided by 8 is 0.