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June 18th, 2011, 08:43 PM   #1
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Number of all non-negative 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.
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June 20th, 2011, 12:52 AM   #2
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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.
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June 20th, 2011, 02:53 AM   #3
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Re:

Quote:
Originally Posted by skipjack
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.
Isn't a = 97, b = 80, c = 1, d = 1 a solution?
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June 20th, 2011, 02:59 AM   #4
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Re: Re:

Quote:
Originally Posted by mrtwhs
Quote:
Originally Posted by skipjack
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.
Sorry! You are correct. I misread b < 90 to say a < 90
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June 22nd, 2011, 06:04 AM   #5
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Re: Number of all non-negative 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+4-1)C(1997). But the coefficients are a stumbling block for me.
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June 22nd, 2011, 06:42 AM   #6
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Can you find the few individual solutions when c = 9 and d = 1?
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July 31st, 2011, 06:28 AM   #7
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Re: Number of all non-negative solutions.

Its a diophantine equation with 4 variables. Find the general formula for each variable and then restrict them to nonnegative values.
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July 31st, 2011, 07:06 AM   #8
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Re: Number of all non-negative 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 non-negative.
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