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When a positive integer \(n\) is divided by \(5\), the remainder is \(2\). What is the remainder when \(n\) is divided by \(3\)?

1) \(n\) is divisible by \(2\) 2) When \(n\) is divided by \(15\), the remainder is \(2\).

Target question:What is the remainder when n is divided by 3?

Given: When positive integer n is divided by 5, the remainder is 2 ----ASIDE---------------------- When it comes to remainders, we have a nice rule that says: If N divided by D leaves remainder R, then the possible values of N are R, R+D, R+2D, R+3D,. . . etc. For example, if k divided by 5 leaves a remainder of 1, then the possible values of k are: 1, 1+5, 1+(2)(5), 1+(3)(5), 1+(4)(5), . . . etc. ----------------------------------- So, from the given information, we can conclude that some possible values of n are: 2, 7, 12, 17, 22, 27, 32, 37, etc

Statement 1: n is divisible by 2 When we examine our list of possible n-values (2, 7, 12, 17, 22, 27, 32, 37, ... ), we see that n could equal 2, 12, 22, 32, 42, etc Let's test two of these possible n-values: Case a: n = 2. In this case, 2 divided by 3 equals 0 with remainder 2. So, the answer to the target question is the remainder is 2 Case b: n = 12. In this case, 12 divided by 3 equals 4 with remainder 0. So, the answer to the target question is the remainder is 0 Since we cannot answer the target question with certainty, statement 1 is NOT SUFFICIENT

Statement 2: When n is divided by 15, the remainder is 2 In other words, n is 2 greater than some multiple of 15 We can write: n = 15k + 2, where k is some integer. IMPORTANT: notice we can take n = 15k + 2 and rewrite it as n = 3(5k) + 2 Notice that 3(5k) is a multiple of 3, which means 3(5k) + 2 is 2 MORE than some multiple of 3 This means that, when we divide n by 3, the remainder is 2 So, the answer to the target question is the remainder is 2 Since we can answer the target question with certainty, statement 2 is SUFFICIENT

Forget conventional ways of solving math questions. For DS problems, the VA (Variable Approach) method is the quickest and easiest way to find the answer without actually solving the problem. Remember that equal numbers of variables and independent equations ensure a solution.

We have 1 variable (n) and 1 equation. So, D is most likely to be the answer, and we should consider each of the conditions on its own first.

Plugging-in numbers is the suggested approach to remainder questions.

Condition 1) The possible values of n are \(n = 2, 4, 6, 8, …\) When these are divided by \(3\), the remainders are \(0, 1\) and \(2\). Since the answer is not unique, condition 1) is not sufficient.

Condition 2) The possible values of n are \(n = 17, 32, 47, 62, …\) When these are divided by \(3\), the remainder is always \(2\). Since the answer is unique, condition 2) is sufficient.

Therefore, B is the answer.

Answer: B

If the original condition includes “1 variable”, or “2 variables and 1 equation”, or “3 variables and 2 equations” etc., one more equation is required to answer the question. If each of conditions 1) and 2) provide an additional equation, there is a 59% chance that D is the answer, a 38% chance that A or B is the answer, and a 3% chance that the answer is C or E. Thus, answer D (conditions 1) and 2), when applied separately, are sufficient to answer the question) is most likely, but there may be cases where the answer is A,B,C or E.
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