Solution:
It is given that the remainder when 10p - q is divided by 3 is R.
Consider first statement (1) alone.
p = 7.
Or 10p - q is 70 - q . If q is 1 , 70 - q is 69 and R is 0 because 69 is divisible by 3.
If q = 2, 70 - q is 68 and R is 2.
We are not getting a definite value of R.
So (1) alone is not sufficient to answer the question.

Next consider (2) alone.
q = 4.
10p - q = 10p - 4.
If p is 1, 10p - 4 = 6 and R is 0.
If p is 2, 10p - 4 = 16 and R is 1.
Again we do not get a definite value of R.

Next combine both the statements together and check.
Or 10p - q is 10*7 - 4 = 66 and so R is 0.

The correct answer is (C) .

pradeepkaushal9518 wrote:If p, q, and R are non-negative integers such that the remainder when 10p - q is divided by 3 is equal to R, what is the value of R?

p = 7
q = 4

I'm not sure why the question is phrased in such an awkward way (there is no need to introduce the letter 'R' here), but is it possible the question instead asks for the remainder when 10^p - q is divided by 3? As written, the question isn't particularly interesting; it's very clear you can answer it with both statements since you then know the exact value of 10p - q, and by testing examples as in the post above, it's easy to rule out each statement alone.

If instead the question asks about 10^p - q, the answer would be B, since 10^p - 4 is always divisible by 3 (it will be a number like 99..996, so is equal to 3*(33...332)), and the remainder will be zero when you divide by 3. Knowing p isn't useful at all, so Statement 1 isn't helpful.

(there's one technicality I ignored in looking at Statement 2, though it does not affect the answer: the question allows p to be zero, but even in that case, 10^0 - 4 = -3 is a multiple of 3. Real GMAT questions typically rule out the possibility of a number being negative if you need to find a remainder, however)