Let S be the smallest positive multiple of 15 that comprises exactly 3k digits with k โ€˜0โ€™s, k โ€˜3โ€™s and k โ€˜8โ€™s. Find the remainder when S is divided by 8.


Answer:

0

Step by Step Explanation:
  1. 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 โ€˜0โ€™s, kโ€˜3โ€™s, and kโ€˜8โ€™s.
    Observe that S must end with 0 as it is a multiple of 5.
  2. The sum of all the digits of S=kร—0+kร—3+kร—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 3โ€˜0โ€™s, 3โ€˜3โ€™s, and 3โ€˜8โ€™s in S.
    โŸนS=300338880
  3. The remainder when S is divided by 8 = Remainder of (Last 3 digits of Sรท8)
    = Remainder of (880รท8)
    =0
         
  4. Hence, the remainder when S is divided by 8 is 0.

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