n is a positive integer. What is the remainder when n is div

[Max@Math Revolution]

n is a positive integer. What is the remainder when n is divided by 3?

1) n^2 has remainder 1 when it is divided by 3
2) n has remainder 7 when it is divided by 9

[jpcameron17]

The easiest way to deal with questions that deal with remainders is to create a list of values that satisfy the remainder condition described. For example, if a question states that the remainder when x is divided by 5 is 3, then create a list of the possible values of x: x = 3, 8, 13, 18, 23, 28,... etc.

1) n^2 has remainder 1 when it is divided by 3
This means that the values of n are n = 1, n = 2, n =4, n = 5, n =49 etc, since only these values have a remainder of 1 when divided by 3. In every single one of these scenarios, the remainder is always 1 when n is divided by 3, therefore Statement (1) is sufficient.

2) n has remainder 7 when it is divided by 9
This means that n = 7, n = 16, n = 25, n = 34, n = 43 etc. The remainder is again always equal to 1 in each of these scenarios when n is divided by 3, thus making the statement sufficient.

The answer is therefore (D).

[Max@Math Revolution]

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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.

Since we have 1 variable (n) and 0 equations, D is most likely to be the answer. So, we should consider each condition on its own first.

Condition 1)
n could be any of the integers 1, 2, 4, 5, 7, 8, ...
If n is one of 1, 4, 7, then n has a remainder 1 when it is divided by 3.
If n is one of 2, 5, 8, then n has a remainder 2 when it is divided by 3.
Thus, condition 1) is not sufficient, since it does not yield a unique solution.

Condition 2)
n = 9k +7 can be expressed as n = 9k + 7 = 9k + 6 + 1 = 3(3k+2)+1. Therefore, n has remainder 1 when it is divided by 3.
Thus, condition 2) is sufficient.

Therefore, B is the answer.
Answer: B