Let SS be the smallest positive multiple of 1515 that comprises exactly 3k3k digits with k ‘0’s, k ‘3’s and k ‘8’s. Find the remainder when SS is divided by 88.


Answer:

00

Step by Step Explanation:
  1. If a number is a multiple of 1515, it is a multiple of 33 and 55 both.
    We are given that SS is the smallest positive multiple of 1515 which comprises exactly 3k3k digits. Also, SS has kk00ss, kk33s,s, and kk88s.s.
    Observe that SS must end with 00 as it is a multiple of 55.
  2. The sum of all the digits of S=k×0+k×3+k×8=3k+8k=11kS=k×0+k×3+k×8=3k+8k=11k
    Since SS is a multiple of 33, the sum of all its digits must be a multiple of 33.
    The smallest value of kk such that 11k11k is a multiple of 33 is 33. Therefore, there are 3300ss, 3333s,s, and 3388ss in SS.
    S=300338880S=300338880
  3. The remainder when SS is divided by 88 == Remainder of (Last 33 digits of S÷8S÷8)
    == Remainder of (880÷8880÷8)
    =0=0
         
  4. Hence, the remainder when SS is divided by 88 is 00.

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