The positive integers p, q, and r each have the same remainder when

Question

The positive integers p, q, and r each have the same remainder when divided by 7. What is the value of q?

(1) p - r =q

(2) 30≤ q≤ 40

NehaKalani · Answer

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gmatophobia · Answer

Let 
 p = 7x_1 + z 
 q = 7x_2 + z 
 z = 7x_3 + z

Statement 1

(1) p - r =q

7(x_1 - x_3) = 7x_2 + z

z = 7(x_1 - x_3- x_2)

Therefore z = 0.

We don't know the value of q.

Statement 2

(2) 30≤ q≤ 40

Not sufficient, as there can be multiple values of q.

Combined

From statement 1, we know that q is a multiple of 7, and from statement 2, we know that p lies between 30 and 40. Hence q = 35

Sufficient.

Option C

Rohan271 · Answer

How did you deduce from statement 1 that z=0 and that q is a multiple of 7?

siddhantvarma · Answer

Rohan271  I'll try to help here.

We have:
p = 7a + k
q = 7b + k
r = 7c + k

From this information alone we can deduce the following:
p - q = 7(a-b)
q - r = 7(b-c)
r - p = 7(c-a) => p - r = 7(a-c)

What do the above expressions tell us? All three integers, p-q, q-r and p-r are multiples of 7. If I can express an integer as any number multiplied by 7, that means the number itself can be divided by 7, leaving a remainder 0, right?
For ex: 35 = 7*5 (Remainder is 0)

Anyway, now coming back to the question.

Statement 1: p - r =q 
We know that p - r is a multiple of 7 and with statement 1 we know that q is also a multiple of 7, since it's value is same as p-r = 7(a-c). Great, so now we know q is a multiple of 7, but which multiple of 7, we still don't know that yet.   Statement 1 is not sufficient => (A) and (D) are out.

Statement 2: 30≤ q≤ 40 
Knowing that q is between 30 and 40 also doesn't tell us anything on it's own. q could be any number.   Statement 2 is not sufficient => (B) is out.

Let's now combine Statements (1) and (2) together;
We now know that q is a multiple of 7 between 30 and 40, which tells us that q is 35. No other value of q satisfies these conditions. Hence, both statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.   Answer is (C).

The positive integers p, q, and r each have the same remainder when

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