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June 18th, 2011, 08:43 PM  #1 
Newbie Joined: Jun 2011 Posts: 5 Thanks: 0  Number of all nonnegative solutions.
This problem appeared in the British Mathematical Olympiad in 1997. Find total number of positive integral solutions for this equation. a+10b+100c+1000d=1997 The question asks for all combinations of values of a,b,c and d. I'd appreciate help on this. Thanks in advance. 
June 20th, 2011, 12:52 AM  #2 
Global Moderator Joined: Dec 2006 Posts: 20,931 Thanks: 2205 
Clearly the last digit of a is 7, b < 90, c < 10, and d = 1. Have you tried to find the total when, for example, c = 9? Do that, then try c = 8, etc.

June 20th, 2011, 02:53 AM  #3  
Senior Member Joined: Feb 2010 Posts: 711 Thanks: 147  Re: Quote:
 
June 20th, 2011, 02:59 AM  #4  
Senior Member Joined: Feb 2010 Posts: 711 Thanks: 147  Re: Re: Quote:
 
June 22nd, 2011, 06:04 AM  #5 
Newbie Joined: Jun 2011 Posts: 5 Thanks: 0  Re: Number of all nonnegative solutions.
The problem is not finding individual solutions of the equation. The problem lies in finding the NUMBER OF ALL POSSIBLE SOLUTIONS. If there were no coefficients i.e a+b+c+d=1997 the total number of all integral nonnegative solutions can be found through (1997+41)C(1997). But the coefficients are a stumbling block for me.

June 22nd, 2011, 06:42 AM  #6 
Global Moderator Joined: Dec 2006 Posts: 20,931 Thanks: 2205 
Can you find the few individual solutions when c = 9 and d = 1?

July 31st, 2011, 06:28 AM  #7 
Newbie Joined: Jun 2011 Posts: 5 Thanks: 0  Re: Number of all nonnegative solutions.
Its a diophantine equation with 4 variables. Find the general formula for each variable and then restrict them to nonnegative values.

July 31st, 2011, 07:06 AM  #8 
Senior Member Joined: Jul 2010 From: St. Augustine, FL., U.S.A.'s oldest city Posts: 12,211 Thanks: 521 Math Focus: Calculus/ODEs  Re: Number of all nonnegative solutions.
With d = 1, the problem reduces to: a+10b+100c = 997 For each c, where 1 ? c < 10, there are 99  10c solutions, thus the total number of solutions S is: I used the wording of the original problem statement where the coefficients are positive, rather than nonnegative. 

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