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What is the reminder when n^2 is divided by 4?

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by Max@Math Revolution » Thu Mar 15, 2018 1:00 am
[GMAT math practice question]

What is the reminder when n^2 is divided by 4?

1) When n is divided by 2, the reminder is 1.
2) When n is divided by 3, the reminder is 1.
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Source: — Data Sufficiency |
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by Vincen » Fri Mar 16, 2018 6:27 am
Max@Math Revolution wrote:[GMAT math practice question]

What is the reminder when n^2 is divided by 4?

1) When n is divided by 2, the reminder is 1.
2) When n is divided by 3, the reminder is 1.
This is how I solve it, I would like to know if there is a better way to do it.

If we use statement (1) then we have that n must be odd, therefore n=1,3,5,7,....

Then n^2=1,9,25,49,81,... If we divide n^2 by 4 we get:
1= 4*0 + 1 (remainder 1).
9= 4*2 + 1 (remainder 1).
25= 4*6 + 1 (remainder 1).
49= 4*12 + 1 (remainder 1).
81= 4*20 + 1 (remainder 1).
.....

Always the remainder is 1. SUFFICIENT.

If we use statement (2) then we have that n=1, 4, 7, 10, 13, 16, 19, 22, 25, ....

Then n^2=1, 16, 49, 100, 169, ....

If we divide n^2 by 4 we get:

1= 4*0 + 1 (remainder 1).
16= 4*4 + 0 (remainder 0).
49= 4*12 + 1 (remainder 1).
100= 4*25 + 0 (remainder 0).

Then we cannot determine the remainder. INSUFFICIENT.

Therefore, the correct answer is A.
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by Max@Math Revolution » Sun Mar 18, 2018 10:07 pm
=>
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 of the conditions on their own first.

Condition 1)
Plugging-in numbers is suggested for remainder questions.
The integers which have a remainder of 1 when divided by 2 are odd.
So,
n: 1, 3, 5, 7, 9, ...
and
n^2: 1, 9, 25, 49, 81, ...
Each value of n^2 has a remainder of 1 when it is divided by 4.
So, condition 1) is sufficient.

Condition 2)
When n = 1, n^2= 1 has a remainder of 1 when it is divided by 4.
When n = 4, n^2=16 has a remainder of 0 when it is divided by 4.
Since we don't have a unique answer, condition 2) is not sufficient.

Therefore, A is the answer.

Answer: A

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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