(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)?

10^n is expressible as the product of

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

in [(n+1)^2/2] ways.

There are (n+1)^2 ordered pairs of (x,y) where 10^n=2^x * 5^y
Each must be joined to one other.  Except where x and y are both n/2.

For 12^n the number is [(2n+1)(n+1)/2]

Posted by Jer on 2011-03-21 14:12:28