Guys,

As you know in some GMAT problems, GMAC loves placing implicit conditions on variables we are working with. The most notable conditions are that the variables be both positive and non-negative (either 0 or positive). For example ( made this question up):

John purchased x amount of apples and y pears and paid $19, how many apples and pears did John buy?
1) Each apple costs $3 and pears $4
2) 6x+5y=12

Now from statement 1) we can say 3x+4y=19,
Now from this simple statement, we know x and y have to be nonnengative and integer (thats an implicit condition we can't buy a negative amounts of apples and pears and we cant buy half an apple). But this statement is sufficient because only one set of values satisfies this equality and that is x=1 and y=4. We don't need the typical two equations, two variables to solve this problem.

However if the question had asked:

John purchased an x amount of apples and y pears and paid $13, how many apples and pears did John buy?
1) Apples cost $3 and pears $4
2) Statement 2 (just ignore)

Statement 1 would be insufficient as {x=3 and y=1} or {x=1 or y=3}.

My question is there a way of determining that ax+by=c (If x and y are both non-negative integers, a,b,c are constants.) without trial and error by plugging? There just has to be a more judious method for determining whether equation 3x+4y=19 or 3x+4y=13 has more than one solution set. If we factor the number on the RHS can that tell us something?

Bunuel, please help. For example, here is a problem you recently did that shows the problem.

1. A citrus fruit grower receives $15 for each crate of oranges shipped and $18 for each crate of grapefruit shipped. How many crates of oranges did the grower ship last week?

a. Last week the number of crates of oranges that the grower shipped was 20 more than twice the number of crates of grapefruit shipped.
b. Last week the grower received a total of $38,700 from the crates of oranges and grapefruit shipped.


A citrus fruit grower receives $15 for each crate of oranges shipped and $18 for each crate of grapefruit shipped. How many crates of oranges did the grower ship last week?

Let x be the # of oranges and y the # of grapefruits. Note that x and y must be an integers. Q: x=?

(1) Last week the number of crates of oranges that the grower shipped was 20 more than twice the number of crates of grapefruit shipped --> x=2y+20. Not sufficient to calculate x

(2) Last week the grower received a total of $38,700 from the crates of oranges and grapefruit shipped --> 15x+18y=38700 --> 5x+6y=12900. Multiple values are possible, for istance: x=180 and y=2000 OR x=60 and y=2100.

(1)+(2) Two unknowns, two different linear equations --> We can calculate unique value of x. Sufficient.

Answer: C.