(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)?

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