April 21st, 2010, 06:54 PM  #1 
Senior Member Joined: Apr 2009 Posts: 106 Thanks: 0  Counting Problem
How many ways can 1,000,000 be expressed as a product of two natural numbers? (View 100*10,000 as different from 10,000*100.) As a product of three natural numbers?

April 21st, 2010, 07:32 PM  #2 
Senior Member Joined: Feb 2009 From: Adelaide, Australia Posts: 1,519 Thanks: 3  Re: Counting Problem
1,000,000 = 2?5?, so any divisor must be of the form where x and y are in {0,1,2,3,4,5,6}. If it is a product of two numbers, the first determines the second, so you could write down the answer straight away. See if you can spot the pattern for three numbers (pick a number for the first and look at possibilities for the other two). 
May 4th, 2010, 10:21 AM  #3 
Senior Member Joined: Apr 2009 Posts: 106 Thanks: 0  Re: Counting Problem
I can see how to find the solution for the first part of this problem (49). I am still having trouble finding the solution to the second part of this problem.

May 4th, 2010, 11:55 AM  #4 
Global Moderator Joined: Nov 2006 From: UTC 5 Posts: 16,046 Thanks: 937 Math Focus: Number theory, computational mathematics, combinatorics, FOM, symbolic logic, TCS, algorithms  Re: Counting Problem
You can reexpress the first part of the problem additively: How many ways can you choose nonnegative a,b,c,d with a+b = 5 and c+d = 5? The second part, so expressed, is: How many ways can you choose nonnegative a,b,c,d,e,f with a+b+c = 5 and d+e+f = 5? In both cases you'll have to decide how to handle repetition. 

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