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 Numerous Numeral Enumeration (Posted on 2011-03-20)
(A) Determine the total number of ways in which 102010(base ten) is expressible as the product of:

(I) Four distinct positive integers arranged in increasing order of magnitude.

(II) Five distinct positive integers arranged in increasing order of magnitude.

(III) Six distinct positive integers arranged in increasing order of magnitude.

(B) What are the respective answers to each of (I), (II) and (III) in part-(A), if 102010(base ten) is replaced by 102010(base 12)?

 No Solution Yet Submitted by K Sengupta Rating: 4.5000 (2 votes)

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 Initial thoughts. | Comment 3 of 10 |
As long as they are distinct, the numbers will order themselves.  That seems a bit of a red herring.

The problem then is to form 4 (or 5 or 6) ordered pairs (x1,y1), (x2,y2)... such that none of them are equal and both x1+x2+...=y1+y2+...=2010

There seems to be a few ways to simplify the problem to get a feel for it:  Decrease 4, decrease 2010, or choose a prime base.

Part (B) is just 12^3468 = 2^6936*3^3468 in base 10 so hopefully a very general formula can be found.

 Posted by Jer on 2011-03-21 01:00:07

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