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Which multiple of 9? (Posted on 2010-09-17) Difficulty: 3 of 5
It is a well known fact that if you permute the digits of a number the difference will be a multiple of 9.

Define the sequence D, where D(n) is the smallest positive value that can be increased by 9n through a permutation of its digits. No leading zeroes are allowed so the first term is D(1)=12 not 10

1) Find the next 14 terms of D.

2) Note D(8) is the greatest n with two digits. What is the greatest n with 3, 4, 5, ... digits?

3) There are some numbers a, b such that a≠b but D(a)=D(b). Prove there are infinitely many such pairs.

4) Sometimes D(n)>9n and sometimes D(n)<9n. Prove that both cases happen an infinity of times.
5) Are there any values of n such that D(n)=9n?

No Solution Yet Submitted by Jer    
Rating: 4.0000 (1 votes)

Comments: ( Back to comment list | You must be logged in to post comments.)
Part 5 Comment 8 of 8 |
This is hard one.  No one has taken a stab at at.  I will point out that although I have found a number equal to its own permuted increase, I dont know if this number is optimal in either sense.
  Posted by Jer on 2010-09-26 21:11:14
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