# Given a set of numbers can you derive the values in a set using division and addition of the products of division?

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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I'm afraid I don't understand what you want them to do. Can you either describe it more formally or give further examples? Is it a coincidence that all the summands in your example are the same ($10$), or is that required? If so, all the summands would have to be $1$ in the other two examples, since those numbers don't have any other common divisors. –  joriki Aug 2 '12 at 2:59
Have you encountered the coin problem, by any chance? –  Ｊ. Ｍ. Aug 2 '12 at 5:58
@joriki Apologies I don't know how to describe it more formally; the 10 was just a coincidence. I've added another example to the question for you - let me know if you'd like more. –  glenstorey Aug 2 '12 at 7:03
@J.M. the coin problem was really helpful, so were the McNugget numbers - I think that's the sort of area I need to look into. –  glenstorey Aug 2 '12 at 7:05
@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? –  joriki Aug 2 '12 at 7:06

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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Thanks Gerry that's really helpful. Is there a way to test if a number is relatively prime? Are there tools or methods to identify if a set is relatively prime or not? –  glenstorey Aug 2 '12 at 5:26
@glen, you can use the Euclidean algorithm to check if two numbers are relatively prime. –  Ｊ. Ｍ. Aug 2 '12 at 5:56