## 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?

#### 1 則留言:

Pop 提到...

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.