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When a positive integer n is divided by 3, what is the remainder?

1) When n is divided by 5, the remainder is 2 2) When n is divided by 6, the remainder is 2

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

Statement 1: When 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 information in statement 1, some possible values of n are: 2, 7, 12, 17, 22, 27, ... Let's TEST a couple of values: Case a: If n = 2, then the answer to the target question is the remainder is 2 when n is divided by 3 Case b: If n = 7, then the answer to the target question is the remainder is 1 when n is divided by 3 Since we cannot answer the target question with certainty, statement 1 is NOT SUFFICIENT

Statement 2: When n is divided by 6, the remainder is 2 -----ASIDE--------------------------------------- There's a nice rule that says, "If N divided by D equals Q with remainder R, then N = DQ + R" For example, since 17 divided by 5 equals 3 with remainder 2, then we can write 17 = (5)(3) + 2 Likewise, since 53 divided by 10 equals 5 with remainder 3, then we can write 53 = (10)(5) + 3 ---------------------------------------------------- Let's take statement 2 and rewrite it as follows: When n is divided by 6, the quotient is k and the remainder is 2 From the above rule, we can write: n = 6k + 2 We can rewrite the right side as: n = 3(2k) + 2 We can see that 3(2k) is a multiple of 3 This means 3(2k) + 2 is 2 greater than a multiple of 3 So, when we divide 3(2k) + 2 by 3 the remainder must be 2 This means the answer to the target question is the remainder is 2 when n is divided by 3 Since we can answer the target question with certainty, statement 2 is SUFFICIENT

Re: When a positive integer n is divided by 3, what is the remainder?
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05 Aug 2018, 19:04

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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.

The first step of the VA (Variable Approach) method is to modify the original condition and the question. We then recheck the question.

Since the divisor (6) of condition 2) is a multiple of the divisor (3) of the question, condition 2) is sufficient.

When we encounter questions related to remainders, plugging in numbers is suggested.

Condition 1) If n = 2, then the remainder when n is divided by 3 is 2. If n = 7, then the remainder when n is divided by 3 is 1. Since we don’t have a unique solution, condition 1) is not sufficient.

Condition 2) The possible values of n are n = 2, 8, 14, 20, … All of these values have a remainder of 2 when they are divided by 3. Thus, condition 2) is sufficient.