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

Can n be expressed as the difference of 2 prime numbers?

1) (n-17)(n-21) = 0
2) (n-15)(n-17)=0

$$n\,\,\mathop = \limits^? \,\,{p_1} - {p_2}\,\,\,\,\left( {{p_1} \,,\, {p_2}\,\,{\rm{primes}}} \right)$$
$$\left( 1 \right)\,\,\,n = 17\,\,\,{\rm{or}}\,\,\,n = 21\,\,\,\,\left\{ \matrix{
\,n = 17 = 19 - 2\,\,\,\,\,\,\, \Rightarrow \,\,\,\left\langle {{\rm{YES}}} \right\rangle \hfill \cr
\,n = 21 = 23 - 2\,\,\,\,\,\,\, \Rightarrow \,\,\,\left\langle {{\rm{YES}}} \right\rangle \,\, \hfill \cr} \right.\,\,\,\,\,\,\,\,\,\,\, \Rightarrow \,\,\,\,\,\,\,\,\left\langle {{\rm{YES}}} \right\rangle $$
$$\left( 2 \right)\,\,\,n = 17\,\,{\rm{or}}\,\,\,n = 15\,\,\,\,\left\{ \matrix{
\,n = 17 = 19 - 2\,\,\,\,\,\,\, \Rightarrow \,\,\,\left\langle {{\rm{YES}}} \right\rangle \hfill \cr
\,n = 15 = 17 - 2\,\,\,\,\,\,\, \Rightarrow \,\,\,\left\langle {{\rm{YES}}} \right\rangle \,\, \hfill \cr} \right.\,\,\,\,\,\,\,\,\,\,\, \Rightarrow \,\,\,\,\,\,\,\,\left\langle {{\rm{YES}}} \right\rangle \,\,$$

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

Regards,
Fabio.

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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-17)(n-21) = 0 is equivalent to the statement n = 17 or n =21
If n = 17, then 17 = 19 - 2 is a difference of two prime numbers and the answer is 'yes'.
If n = 21, then 21 = 23 - 2 is a difference of two prime numbers and the answer is 'yes'.
Since it gives a unique answer, condition 1) is sufficient.

Condition 2)
(n-15)(n-17) = 0 is equivalent to the statement n = 15 or n = 17
If n = 15, then 15 = 17 - 2 is a difference of two prime numbers and the answer is 'yes'.
If n = 17, then 17 = 19 - 2 is a difference of two prime numbers and the answer is 'yes'.
Since it gives a unique answer, condition 2) is sufficient.

Therefore, D is the answer.
Answer: D

In cases where 3 or more additional equations are required, such as for original conditions with "3 variables", or "4 variables and 1 equation", or "5 variables and 2 equations", conditions 1) and 2) usually supply only one additional equation. Therefore, there is an 80% chance that E is the answer, a 15% chance that C is the answer, and a 5% chance that the answer is A, B or D. Since E (i.e. conditions 1) & 2) are NOT sufficient, when taken together) is most likely to be the answer, it is generally most efficient to begin by checking the sufficiency of conditions 1) and 2), when taken together. Obviously, there may be occasions on which the answer is A, B, C or D.