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June 4th, 2018, 11:05 AM   #1
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Multiple Multiples

Hello,

Pretend I wanted to buy magic pencils. There's only one store that sells them and they are sold in packages of 6, 9, and 15. What is the most straightforward way to figure out if I can buy X number of magic pencils? For example, if I want to buy 4,238 pencils, would that be possible using only the multiples of 6, 9, and 15? What about 88,904 magic pencils? What's the generic way to go about solving this problem? I am most interested in solving this using only elementary school level math.
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June 4th, 2018, 11:14 AM   #2
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Problems like this are known as linear diophantine equations and are a well studied branch of number theory.

Number theory isn't really my thing but it appears there are well known methods for solving systems of these equations.

Elementary school level math it's not, but the individual steps of the algorithm look straightforward enough.

I'd start here.
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June 4th, 2018, 12:01 PM   #3
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Originally Posted by romsek View Post
Number theory isn't really my thing but it appears there are well known methods for solving systems of these equations.
As I understand it there is no general method for solving Diophantine equations.

https://en.wikipedia.org/wiki/Hilbert%27s_tenth_problem
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June 4th, 2018, 01:28 PM   #4
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As I understand it there is no general method for solving Diophantine equations.
I defer to your knowledge. I did one undergrad course in number theory. It was voodoo.
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June 4th, 2018, 01:57 PM   #5
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Originally Posted by mjsilverfly View Post
Hello,

Pretend I wanted to buy magic pencils. There's only one store that sells them and they are sold in packages of 6, 9, and 15. What is the most straightforward way to figure out if I can buy X number of magic pencils? For example, if I want to buy 4,238 pencils, would that be possible using only the multiples of 6, 9, and 15? What about 88,904 magic pencils? What's the generic way to go about solving this problem? I am most interested in solving this using only elementary school level math.
As 6, 9, and 15 are all multiples of 3, the answer in this case is that you can generate any multiple of 3 greater than 6. Since neither 4238 nor 88904 is a multiple of 3, they cannot be done.

I think you may have meant 6, 9, and 20 (not 15). If that is the case then you may want to google "chicken mcnuggets problem". I don't think there is a general procedure to solve these but here is a heuristic that might help. Make a table with six columns:

1 2 3 4 5 [6]
7 8 [9] 10 11 12
13 14 15 16 17 18
19 [20] 21 22 23 24
25 26 27 28 [29] 30
31 32 33 34 35 36
37 38 39 [40] 41 42
43 44 45 46 47 48
[49] 50 51 52 53 54

Every time you find a solution, put it in brackets. As soon as you find a solution in each column, you have a potential solution to the problem. For 6, 9, and 20, I have found solutions 6 = 6+0+0, 9 = 0+9+0, 20 = 0+0+20, 29 = 0+9+20, 40 = 0+0+20+20, and 49 = 0+9+20+20. Any number below a solution is obtainable. So in this case (6,9,20) any number greater than 43 can be purchased.

Last edited by skipjack; June 5th, 2018 at 08:02 PM.
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June 5th, 2018, 03:23 PM   #6
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OK, thank you. I like your heuristic, mrtwhs.
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