Diophantine equation? "McNuggets come in packs of 6, 9,..."

Complex numbers, rational functions, logarithms, sequences and series, matrix operations, etc.
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Joined: Sun Sep 12, 2010 10:18 pm

Diophantine equation? "McNuggets come in packs of 6, 9,..."

Postby Bug » Mon Sep 13, 2010 12:44 am

Wasn't sure which section this went in sorry, my first post!

I am taking a free online intro to computer science course from MIT. it's teaching basic computer programing in the python language. that's not what i'm here for though. while the program is more logic then math, the professor likes to put out assignments want us to write programs to solve math equations.. unfortunately i don't understand the math behind this one at all.
McDonanld McNuggets come in packs of 6, 9, and 20.
6a + 9b + 20c = n.

Problem 1.

Show that it is possible to buy exactly 50, 51, 52, 53, 54, and 55 McNuggets, by finding
solutions to the Diophantine equation. You can solve this in your head, using paper and pencil,
or writing a program. However you chose to solve this problem, list the combinations of 6, 9
and 20 packs of McNuggets you need to buy in order to get each of the exact amounts.

Given that it is possible to buy sets of 50, 51, 52, 53, 54 or 55 McNuggets by combinations of 6,
9 and 20 packs, show that it is possible to buy 56, 57,…, 65 McNuggets. In other words, show
how, given solutions for 50-55, one can derive solutions for 56-65.

Theorem: If it is possible to buy x, x+1,…, x+5 sets of McNuggets, for some x, then it is
possible to buy any number of McNuggets >= x, given that McNuggets come in 6, 9 and 20

Problem 2.

Explain, in English, why this theorem is true.

from there i am required to build two programs, the first seeks out the largest number of McNuggets that cannot be bought in
exact quantity. and the other one builds off of that. I'm not askng about the programs themselves, just the math.
i went with paper and trial and error and found:

(6x5 )+ (9x0) + (20x1) = 50
(6x4) + (9x3) + (20x0) = 51
(6x2) + (9x0) + (20x2) = 52
(6x1) + (9x3) + (20x1) = 53
(6x0) + (9x6) + (20x0) = 54
(6x1) + (9x1) + (20x2) = 55

i am confident i could do the same for 56 through 65 in the same method but it sounds like he wanted me to use the theorem some how.
ok what i am looking for: if someone could restate that theorem in more....beginner friendly words so its easier to understand ( i should note, i have both dyslexia and dyscalculia)? and then some hints on how to 'prove it' and 'use it'?
any help would be great! thanks!

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Joined: Wed Mar 11, 2009 3:28 pm

Re: Diophantine equation?

Postby nicknation » Mon Sep 13, 2010 1:15 am

this is an OLD one! you can find it here:

Posts: 2
Joined: Sun Sep 12, 2010 10:18 pm

Re: Diophantine equation? "McNuggets come in packs of 6, 9,.

Postby Bug » Tue Sep 14, 2010 6:41 pm

ah well that answered Problem 3 which i wasn't really looking for.. but! somehow in the course of reading it everything clicked and i figured out what i wasn't getting. (i find these random bouts of insight happen often with the dyscalculia. ) in either case. i got what i needed so THANK YOU!!!

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