2009年12月24日 星期四

Elementary number theory

Someone say that 167588402882520529579353108764873470755823697 is the smallest positive integer k such that all digit of 1989k are the same.

Do you agree?
於

1 則留言:

  1. Pop2009年12月25日 凌晨3:20

    1989k = 9*13*17*k =r(111...1), where r between 1-9

    Numbers divisible by 13
    alternating sum of three-digit, if it is divisible by 13, then the number is divisible by 13.

    so 111,111 is divisible by 13 as (111)-(111)=0

    Numbers divisible by 17
    alternating sums of 8-digit groups!

    so 1...16-digit...1 is divisible by 17.

    so 1....48-digit...1 is the smallest integer of this form which can be divisible by both 13 and 17.

    and so the smallest positive integer k s.t. all digit of 1989k are the same = 9*13*17*k = (r...48 digit...r)

    if r=1, 48 != 0 mod 9, so not the answer.
    if r=2, 96 !=0 mod 9, so not the answer.
    if r=3, 144 = 0 mod 9, 3...48digit...3 is the required answer!

    the given k = 167588402882520529579353108764873470755823697 have 45 digits,
    16758
    8402882520
    5295793531
    0876487347
    0755823697

    so this k times 1989 will have 48 digits.

    If the question guarantee this k times 1989 would produces a number with all digit are the same, then i can 100% agree.

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