Mathematical principal behind linear equation for DS problem

[kapoor.sam]

On a DS problem, if we are asked whether we can find the values of 2 integers x and y in an equation such as:
5x + 2y = 23
the answer would be insufficient since x and y could take on different values.

However, if we are given an equation such as
7x + 3y = 16
then we can say it is sufficient since the only values that would work for x and y would be 1 and 3.

What is the mathematical principal behind this?
The above examples are very simple but in real GMAT problems the numbers in the equation could be quite large and it won't always be possible to tell intuitively if only a single combination of values works for x and y.

I thought that this has something to do with prime numbers in the equation but don't think that is the case.

Any help would be appreciated from the Quant experts out here.

Thanks:

-Saket

[jnelson0612]

On a DS problem, if we are asked whether we can find the values of 2 integers x and y in an equation such as:
5x + 2y = 23
the answer would be insufficient since x and y could take on different values.

However, if we are given an equation such as
7x + 3y = 16
then we can say it is sufficient since the only values that would work for x and y would be 1 and 3.

What is the mathematical principal behind this?
The above examples are very simple but in real GMAT problems the numbers in the equation could be quite large and it won't always be possible to tell intuitively if only a single combination of values works for x and y.

I thought that this has something to do with prime numbers in the equation but don't think that is the case.

Any help would be appreciated from the Quant experts out here.

Thanks:

-Saket

Hi Saket!

First, I've never seen a Data Sufficiency question ask for two different values. The combination of two values certainly, such as "what is xy?", but I've never seen a question ask "what is x and what is y"?

Also, be careful about your assumptions! What if x=4 and y=-4 in your second equation? Remember that integers can be positives, zero, and negatives.

I'm not fully understanding your question. Frequently by testing numbers you can find two possibilities and prove insufficiency. If that is not an option then usually the GMAT will give you some other piece of information that will be fairly obviously helpful or fairly obviously not helpful. You are right that the particular coefficients may play somewhat of a part, but it really depends on what those are. I'm not seeing the prime coefficients as particularly important, but again, it depends on the problem.

[kapoor.sam]

Thanks Jamie for the response.

[RonPurewal]

i think i know what you're getting at here. there are quite a few data sufficiency problems in which the solutions have to be whole numbers -- usually because of circumstances that arise naturally from a word problem (e.g., you can't have non-whole numbers of boys and girls) -- and you're looking for a shortcut that will get you out of having to test cases.

the problem is that, in general, there is no such shortcut. (in fact, if you could find one, you would become a famous and influential mathematician overnight; many mathematicians have actually written their doctoral theses on the reasons why you can't find algebraic solutions to things like these.)
in general, you're just going to have to get out the shovel and start shoveling the dirt -- in other words, if your solutions are restricted to whole numbers, you're going to have to test cases.

just as an extra nail in the coffin of the idea that there might be some simple algebraic resolution here, consider the following two equations:
5x + 7y = 47
5x + 7y = 48
the first one has two solutions in whole numbers, (8, 1) and (1, 6). the second has only one such solution, (4, 4). there's nothing much different about the coefficients -- in both cases, they are all numbers that have no factors in common other than 1. but the results are different nonetheless.

on the bright side, testing cases really doesn't take long at all -- and it's infallible! if you have a literal list of cases in front of your eyes, it's impossible to fall into the kind of traps that can ensnare you if you are doing algebra.

--

of course, if the number on the right-hand side of the equation is gigantically huge, then it can become obvious that there are multiple solutions.
for instance, if i give you 5x + 7y = 10,000, then it should be clear that you can find a whole lot of different pairs of x and y that will solve that one. (if that's not clear, think about the fact that 5x + 7y will have the same value if you increase x by 7 and decrease y by 5, or vice versa.)