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This is my first question on the Math SE, and it may be a bit simple. Apologies if my question reveals my misunderstanding of your field or the universe in general. Me Programmer.

I am working on a maths game to teach kids the use of multiplication, division and logic. Kids are given a sum of a set, and the values in a set. They need to show that they can work out the numbers in the set by dividing the sum into smaller parts (dividing by 1-10) and adding the product of that division to other products of that division. They can only use whole numbers.

For a really simple example, with a set of 30, 30 and 40 (and a sum of 100), can be solved by:

100/10          (split 100 into ten parts) 10+10+10        (add three lots of 10, etc.) 10+10+10         10+10+10+10      

Another example:

With a set of 18, 18, 12, and 6 and a sum of 54:

54/2    = 18 with 18 left over r18     = 18 r18/3*2 = 12 with 6 left over r6      = 6 

But I can't see a way to solve this with a Fibonacci sequence:

1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144 with the sum 376 

Or a set of prime numbers:

2, 3, 5, 7, 11, 13, 17, 19, 23 with the sum 100 

Given the sum of a set of numbers, is there a relationship between the sum of the set and the numbers in the set (other than the obvious) that I can exploit to create problems for the kids to solve? Is there a way to test whether a set is solvable in this game, or a formula I should be using to test/generate the problems?

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    @glenstorey: The example you added has only made things even more mysterious to me. Could you try to make more of an effort to give a more formal description of the task?2012-08-02

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The thing about prime numbers is that you can't divide them by whole numbers and get whole numbers, except trivially. That is, you can split 100 into 100 parts, and then do
$2=1+1$
$3=1+1+1$
$5=1+1+1+1+1$
etc., but that's really all you can do with primes.

More generally, if you have two numbers that are relatively prime, say, $6+49=55$, all you can do is split the 55 into 55 parts and do
$6=1+1+1+1+1+1$
$49=1+1+\dots+1$

As soon as one of your numbers is 1 (as in the Fibonaccis), you're guaranteed to have numbers that are relatively prime, and you're reduced to splitting everything into ones.

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    @glen, you can use the Euclidean algorithm to check i$f$ two numbers are relatively prime.2012-08-02