Data Sufficiency

[Math Revolution GMAT math practice question]

If n is an integer, is n(n+2) divisible by 8?

1) n is an even number.
2) n is a multiple of 4.

Max@Math Revolution wrote:[Math Revolution GMAT math practice question]

If n is an integer, is n(n+2) divisible by 8?

1) n is an even number.
2) n is a multiple of 4.

$$n\,\,{\mathop{\rm int}} $$
$${{n\left( {n + 2} \right)} \over 8}\,\,\mathop = \limits^? \,\,{\mathop{\rm int}} $$

$$\left( 1 \right)\,\,n = 2M,\,\,M\,\,{\mathop{\rm int}} $$
$${{n\left( {n + 2} \right)} \over 8} = {{2M \cdot 2 \cdot \left( {M + 1} \right)} \over 8} = {{M\left( {M + 1} \right)} \over 2}\mathop = \limits^{\left( * \right)} \,\,{\mathop{\rm int}} \,\,\,\,\, \Rightarrow \,\,\,\,\,{\rm{SUFF}}.$$

(*) The product of two consecutive integers is even (odd*even or even*odd), hence the integer M(M+1) has (at least) one factor 2 in its decomposition...

$$\left( 2 \right)\,\,n = 4J\,\,,\,\,J\,\,{\mathop{\rm int}} $$
$${{n\left( {n + 2} \right)} \over 8} = {{4J \cdot 2 \cdot \left( {2J + 1} \right)} \over 8} = J \cdot \left( {2J + 1} \right) = {\mathop{\rm int}} \cdot {\mathop{\rm int}} = {\mathop{\rm int}} \,\,\,\,\, \Rightarrow \,\,\,\,\,{\rm{SUFF}}.$$


This solution follows the notations and rationale taught in the GMATH method.

Regards,
fskilnik.

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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 of the conditions on their own first.

Condition 1)
If n is an even integer, then n and n + 2 are two consecutive even integers.
Products of two consecutive even integers are multiples of 8 since one of them must be a multiple of 4, and the other a multiple of 2.
Condition 1) is sufficient.

Condition 2)
Since n is a multiple of 4, n + 2 is an even integer.
Thus, n(n+2) is a multiple of 8.
Condition 2) is sufficient.

Therefore, D is the answer.

Answer: D

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.