Data Sufficiency

If n is a positive integer and r is the remainder when 4+7n is divided by 3 what is the value of r ?

(1) n+1 is divisible by 3
(2) n >20

rommysingh wrote:If n is a positive integer and r is the remainder when 4+7n is divided by 3 what is the value of r ?

(1) n+1 is divisible by 3
(2) n >20

Looking at Statement 1, you can see that any multiple of n + 1 will also be divisible by 3. So 2n + 2 is divisible by 3, 3n + 3 is divisible by 3, and so on.

So when you see the 7n in the question you can think 7n + 7 is divisible by 3. Okay, so what about 7n + 4? Well 7n + 4 = (7n + 7) - 3. So if 7n + 7 is divisible by 3, then (7n + 7) - 3 = 7n + 4 is also divisible by 3, because 7n + 4 is 3 less than a multiple of 3.

Given that 7n + 4 is divisible by 3, dividing 7n + 4, or 4 + 7n, by 3 generates a remainder of 0.

So Statement 1 is sufficient.

Statement 2 might have been part of a C trap, but otherwise looks totally useless. Still I guess it makes sense to plug in some numbers to confirm that.

If n = 30, 4 + 7n = 214. 210 is a multiple of 3. So dividing 214 by 3 generates a remainder of 1.

If n = 40, 4 + 7n = 284. 282 is a multiple of 3, as determined by adding 2 + 8 + 2 and getting 12, which is a multiple of 3. So dividing 284 by 3 generates a remainder of 2.

So using Statement 2 one can generate various remainders, and Statement 2 is insufficient.

Maybe even checking Statement 2 is a waste of time as it seems pretty obvious that it is insufficient, but maybe it makes sense to be careful, as when one is under the pressure of taking the test one can come to some funny conclusions sometimes.

In any case the answer to this question is A.

Marty/Brent,

In the first statement, you showed why it's sufficient. In the second one, you plugged in numbers. How did you know to plug in numbers? Why didn't you do that for statement 1?

In the first statement, you showed why it's sufficient. In the second one, you plugged in numbers. How did you know to plug in numbers? Why didn't you do that for statement 1?

The strategy you employ is a matter of taste/instinct. You certainly could pick numbers for statement 1.

Say n + 1 = 3. Then n = 2. And 4 + 7n = 4 + 14 = 18, which gives a remainder of 0 when divided by 3.
Say n + 1 = 6. Then n = 5. And 4 + 7n = 4 + 35 = 39, which gives a remainder of 0 when divided by 3.

No matter what you pick, the remainder will be 0, so long as you satisfy statement 1.

Marty and Brent showed some neat approaches that can help you better understand number properties. Statement 2, on the other hand, is very open-ended, so it seemed logical to pick numbers. The best approach is the one that gets you to a correct answer in a reasonable amount of time.

DavidG@VeritasPrep wrote:

In the first statement, you showed why it's sufficient. In the second one, you plugged in numbers. How did you know to plug in numbers? Why didn't you do that for statement 1?

The strategy you employ is a matter of taste/instinct. You certainly could pick numbers for statement 1.

Say n + 1 = 3. Then n = 2. And 4 + 7n = 4 + 14 = 18, which gives a remainder of 0 when divided by 3.
Say n + 1 = 6. Then n = 5. And 4 + 7n = 4 + 35 = 39, which gives a remainder of 0 when divided by 3.

No matter what you pick, the remainder will be 0, so long as you satisfy statement 1.

Marty and Brent showed some neat approaches that can help you better understand number properties. Statement 2, on the other hand, is very open-ended, so it seemed logical to pick numbers. The best approach is the one that gets you to a correct answer in a reasonable amount of time.

Thanks David!

Forget conventional ways of solving math questions. In DS, Variable approach is the easiest and quickest way to find the answer without actually solving the problem. Remember equal number of variables and independent equations ensures a solution.

If n is a positive integer and r is the remainder when 4+7n is divided by 3 what is the value of r ?

(1) n+1 is divisible by 3
(2) n >20

We solve remainder problems by directly substituting in values. Looking at the original condition, we can see that there is only 1 variable (n) We therefore need 1 equation, and 2 equations are given, so there is high chance (D) will be our answer.
From condition 1, n+1=3t(t is a positive integer), So n=3t-1=2,5,8,11,.......(directly substituted) Then 4+7n=18, 39, 60... All these values divided by 3 leave remainders of 0, so this is a sufficient condition.
For condition 2, n=21,22 and 4+7n=151, 158, and 151=3*50+1, so the remainder is 1, but for 158=3*52+2, the remainder is 2, and this is not unique, so this condition is insufficient and therefore the answer is (A).

For cases where we need 1 more equation, such as original conditions with "1 variable", or "2 variables and 1 equation", or "3 variables and 2 equations", we have 1 equation each in both 1) and 2). Therefore, there is 59 % chance that D is the answer, while A or B has 38% chance and C or E has 3% chance. Since D is most likely to be the answer using 1) and 2) separately according to DS definition. Obviously there may be cases where the answer is A, B, C or E.

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