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If m and n are positive integers, is m^2-n^2 divisible by 4?

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by Max@Math Revolution » Tue Apr 09, 2019 11:37 pm

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[GMAT math practice question]

If m and n are positive integers, is m^2-n^2 divisible by 4?

1) m^2+n^2 has remainder 2 when it is divided by 4
2) m*n is an odd integer
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Source: — Data Sufficiency |
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by Max@Math Revolution » Fri Apr 12, 2019 12:27 am
=>

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.

The statement "m^2-n^2 is divisible by 4" means that (m+n)(m-n) is divisible by 4. This is equivalent to the requirement that m and n are either both even integers or both odd integers.

Since condition 2) tells us that both m and n are odd integers, condition 2) is sufficient.

Condition 1)
The square of an odd integer (2a+1)^2 = 4a^2 + 4a + 1 = 4(a^2 + a) + 1 has remainder 1 when it is divided by 4.
The square of an even integer (2b)^2 = 4b^2 has remainder 0 when it is divided by 4.
Thus, if "m^2+n^2 has remainder 2 when it is divided by 4", both m and n must be odd integers.
Condition 1) is sufficient.

Therefore, D is the answer.
Answer: D

FYI, Tip 1) of the VA method states that D is most likely to be the answer if conditions 1) and 2) provide the same information.
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question

by [email protected] » Mon Apr 15, 2019 6:12 am
Can u please clarify the highlighted portion below ?



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.

The statement "m^2-n^2 is divisible by 4" means that (m+n)(m-n) is divisible by 4. This is equivalent to the requirement that m and n are either both even integers or both odd integers.[/color]

Since condition 2) tells us that both m and n are odd integers, condition 2) is sufficient.

Condition 1)
The square of an odd integer (2a+1)^2 = 4a^2 + 4a + 1 = 4(a^2 + a) + 1 has remainder 1 when it is divided by 4.
The square of an even integer (2b)^2 = 4b^2 has remainder 0 when it is divided by 4.
Thus, if "m^2+n^2 has remainder 2 when it is divided by 4", both m and n must be odd integers.
Condition 1) is sufficient.

Therefore, D is the answer.
Answer: D

FYI, Tip 1) of the VA method states that D is most likely to be the answer if conditions 1) and 2) provide the same information.[/quote] $$$$ $$$$ $$$$ $$$$
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reply

by [email protected] » Tue Apr 16, 2019 9:41 pm
Max@Math Revolution wrote:=>

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.

The statement "m^2-n^2 is divisible by 4" means that (m+n)(m-n) is divisible by 4. This is equivalent to the requirement that m and n are either both even integers or both odd integers.

Since condition 2) tells us that both m and n are odd integers, condition 2) is sufficient.

Condition 1)
The square of an odd integer (2a+1)^2 = 4a^2 + 4a + 1 = 4(a^2 + a) + 1 has remainder 1 when it is divided by 4.
The square of an even integer (2b)^2 = 4b^2 has remainder 0 when it is divided by 4.
Thus, if "m^2+n^2 has remainder 2 when it is divided by 4", both m and n must be odd integers.
Condition 1) is sufficient.

Therefore, D is the answer.
Answer: D

FYI, Tip 1) of the VA method states that D is most likely to be the answer if conditions 1) and 2) provide the same information.


Can u please clarify the highlighted portion below ?
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