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).
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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