What is the remainder when the positive integer n is divided by the po

What is the remainder when the positive integer n is divided by the positive integer k, where k>1

(1) \(n=(k+1)^3= k^3 + 3k^2 + 3k + 1=k(k^2+3k+3)+1\) --> first term, \(k(k^2+3k+3)\), is obviously divisible by \(k\) and 1 divide by \(k\) yields the remainder of 1 (as \(k>1\)). Sufficient.

(2) \(k=5\). Know nothing about \(n\), hence insufficient.

Answer: A.
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bunuel's solution is certainly the quickest way to solve. (Of course, it would be (k+1)^3 = k^3 + 3k^2 + 3k + 1, and since each of the "k" terms is divisible by k, the remainder is 1).

But if this deduction/concept didn't jump out at you, you could also solve fairly quickly by picking numbers. If we pick different integer values for k and always end up with the same remainder for n/k, then we can trust that (1) is sufficient. On the other hand, if ever we get different remainder values, we know at once (1) is insufficient:

(1): let k=2. Then, n = (2+1)^3 = 27. 27/2 leaves a remainder of 1.
----let k=3. Then, n = (3+1)^3 = 64. 63 is clearly divisible by 3. Thus, 64/3 leaves a remainder of 1 again.
----let k=4. Then, n = (4+1)^3 = 125. 124 is clearly divisible by 4. Thus, 125/4 leaves a remainder of 1 again.

We've convinced ourselves that (1) is sufficient!

Hi,

Here are my two cents for this question.


if \(\frac{(ax+1)^n}{a}\) remainder is always 1 ----------(I)

eg \(\frac{(37)^{261}}{9}\) = \(\frac{(9*4+1)^{261}}{9}\) in which case remainder is \(\frac{(1)^{261}}{9}\) which is 1 .

or from above we could see it has same form as \(\frac{(ax+1)^n}{a}\) so reminder is 1.

So this question is using the above logic.

So we can say our Answer to this question is A.

More on this you can find at

https://gmatclub.com/forum/using-remain ... l#p2243712
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Plugging in number;
1) Let, k= 2
so, n = 27, and 27/2; remainder is 1
K= 3, n=64; 64/3; remainder is 1
Thus A is sufficient.

2) No information about N is given. ; insufficient

The answer is A.

Forget the conventional way to solve DS questions.

We will solve this DS question using the variable approach.

Remember the relation between the Variable Approach, and Common Mistake Types 3 and 4 (A and B)[Watch lessons on our website to master these approaches and tips]

Step 1: Apply Variable Approach(VA)

Step II: After applying VA, if C is the answer, check whether the question is key questions.

StepIII: If the question is not a key question, choose C as the probable answer, but if the question is a key question, apply CMT 3 and 4 (A or B).

Step IV: If CMT3 or 4 (A or B) is applied, choose either A, B, or D.

Let's apply CMT (2), which says there should be only one answer for the condition to be sufficient. Also, this is an integer question and, therefore, we will have to apply CMT 3 and 4 (A or B).

To master the Variable Approach, visit https://www.mathrevolution.com and check our lessons and proven techniques to score high in DS questions.

Let’s apply the 3 steps suggested previously. [Watch lessons on our website to master these 3 steps]

Step 1 of the Variable Approach: Modifying and rechecking the original condition and the question.

We have to find the remainder when the positive integer n is divided by the positive integer, where k >1.

Second and the third step of Variable Approach: From the original condition, we have 2 variables (n and k). To match the number of variables with the number of equations, we need 2 equations. Since conditions (1) and (2) will provide 1 equation each, C would most likely be the answer.

But we know that this is a key question [Integer question] and if we get an easy C as an answer, we will choose A or B.

Let’s take a look at each condition.

Condition(1) tells us that n = \((k + 1)^3\).

=> \(k^3\) is always divisible by k

=> If k = 2 then \(k^3\) = \(2^3\) = 8. => 8 is divisible by 2.

=> If k = 5 then \(k^3 = 5^3\) = 125. => 125 is divisible by 5.

=> \((k + 1)^3\) = \(k^3 + 3k^2\)+3K+1 [\(k^2\) and 3k is always divisible by k]

Hence \((k+1)^3\) will always give '1' as a remainder when divided by K.

Since the answer is unique, the condition is sufficient by CMT 2.


Condition(2) tells us that K = 5.

=> We have no information on 'n'.

Since the answer is not unique, the condition is not sufficient by CMT 2.

Condition (1) is alone sufficient.

So, A is the correct answer.

Answer: A

1st. n = (k + 1)^3

Remainder = \(\frac{(k + 1)^3}{k}\) = \(\frac{(k + 1)*(k + 1)*(k + 1)}{k}\) = \(\frac{1}{k}\)*\(\frac{1}{k}\)*\(\frac{1}{k}\) =\(\frac{1}{k}\)

Remainder is "1" Thus, Statement 1 is sufficient.

2st. k = 5 - no info about "n" hence, insufficient.

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