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# If n is an integer and n > 1,000^2 - 999^2, n = ?

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Math Revolution GMAT Instructor
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If n is an integer and n > 1,000^2 - 999^2, n = ? [#permalink]

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30 Nov 2017, 00:24
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Difficulty:

45% (medium)

Question Stats:

69% (00:58) correct 31% (01:08) wrong based on 48 sessions

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[GMAT math practice question]

If $$n$$ is an integer and $$n>1,000^2-999^2$$, n = ?

(1) $$n<1,001^2-1,000^2$$

(2) $$n<502^2-500^2$$
[Reveal] Spoiler: OA

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30 Nov 2017, 00:28
Option A

From question stem we can rearrange and deduce n>1999

From st1 we can rearrange and deduce n<2001. Sufficient. It means n=2000

From st2 we can rearrange and deduce n<2004. There are 4 values of n as 2000, 2001, 2002 & 2003. Insufficient.

Hence A

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Kudos [?]: 38 [0], given: 89

Math Revolution GMAT Instructor
Joined: 16 Aug 2015
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Kudos [?]: 3154 [0], given: 0

GPA: 3.82
If n is an integer and n > 1,000^2 - 999^2, n = ? [#permalink]

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03 Dec 2017, 17:33
=>

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, and then recheck the question

We can modify the original condition as follows:

$$n>1,000^2-999^2$$
$$⇔ n > (1000+999)(1000-999)$$
$$⇔ n > 1999$$

Since we have 1 variable (n) and 0 equations, D is most likely to be the answer.

Condition 1)

$$n<1,001^2-1,000^2$$
$$⇔ n < (1001+1000)(1001-1000)$$
$$⇔ n < 2001$$

Since the original condition and condition 1) combine to give $$1999 < n < 2001$$, we must have $$n = 2000$$.
We have a unique solution.
Therefore, condition 1) is sufficient.

Condition 2)

$$n<502^2-500^2$$
$$⇔ n < (502+500)(502-500)$$
$$⇔ n < 1002*2$$
$$⇔ n < 2004$$

Since the original condition and condition 2) combine to give $$1999 < n < 2004$$, $$n = 2000, 2001, 2002, or 2003.$$
We don’t have a unique solution.
So, condition 2) is not sufficient.

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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If n is an integer and n > 1,000^2 - 999^2, n = ?   [#permalink] 03 Dec 2017, 17:33
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