[Math Revolution GMAT math practice question]
If m and n are integers, is m+m^2-n an even number?
1) m is an even number
2) n is an even number
If m and n are integers, is m+m^2-n an even number?
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- Max@Math Revolution
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$$m,n\,\,\,{\rm{ints}}\,\,\left( * \right)$$Max@Math Revolution wrote:[Math Revolution GMAT math practice question]
If m and n are integers, is m+m^2-n an even number?
1) m is an even number
2) n is an even number
$$m + {m^2} - n\,\,\mathop = \limits^? \,\,{\text{even}}\,\,\,\,\,\,\mathop \Leftrightarrow \limits^{\left( {**} \right)} \,\,\,\,\boxed{\,\,n\,\,\mathop = \limits^? \,\,{\text{even}}\,\,}$$
$$\left( ** \right)\,\,m + {m^2} = m\left( {m + 1} \right)\,\,\mathop = \limits^{\left( * \right)} \,\,{\rm{even}}$$
$$\left( 1 \right)\,\,m\,\,{\rm{even}}\,\,\,\left\{ \matrix{
\,{\rm{Take}}\,\,\left( {m,n} \right){\rm{ = }}\left( {{\rm{0}}{\rm{,0}}} \right)\,\,\,\, \Rightarrow \,\,\,\left\langle {{\rm{YES}}} \right\rangle \hfill \cr
\,{\rm{Take}}\,\,\left( {m,n} \right){\rm{ = }}\left( {{\rm{0}}{\rm{,1}}} \right)\,\,\,\, \Rightarrow \,\,\,\left\langle {{\rm{NO}}} \right\rangle \hfill \cr} \right.$$
$$\left( 2 \right)\,\,n\,\,{\rm{even}}\,\,\, \Rightarrow \,\,\,\left\langle {{\rm{YES}}} \right\rangle $$
The correct answer is therefore (B).
This solution follows the notations and rationale taught in the GMATH method.
Regards,
Fabio.
Fabio Skilnik :: GMATH method creator ( Math for the GMAT)
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Given: m and n are integersMax@Math Revolution wrote:[Math Revolution GMAT math practice question]
If m and n are integers, is m + m² - n an even number?
1) m is an even number
2) n is an even number
Target question: Is m + m² - n an even number?
This is a good candidate for rephrasing the target question
Aside: Here's a video with tips on rephrasing the target question: https://www.gmatprepnow.com/module/gmat- ... cy?id=1100
We can take the expression, m + m² - n and rewrite it as m(1 + m) - n
Now notice that m and (1+m) are CONSECUTIVE integers, which means one value, either m or (m+1), is EVEN, and the other value is ODD
So, we can be certain that the product m(1 + m) is EVEN
So, our question becomes Is (some even number) - n even?
In order for this value to be even, we need n to be even
So, we can REPHRASE the target question.....
REPHRASED target question: Is n EVEN?
Statement 1: m is an even number
Since we're given no information about n, we cannot answer the REPHRASED target question
Statement 1 is NOT SUFFICIENT
Statement 2: n is an even number
Perfect! The answer to the REPHRASED target question is YES, n IS even
Since we can answer the target question with certainty, statement 2 is SUFFICIENT
Answer: B
Cheers,
Brent
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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.
The first step of the VA (Variable Approach) method is to modify the original condition and the question. We then recheck the question.
m+m^2-n = m(m+1) - n. Now, m(m+1) is an even number since m(m+1) is the product of two consecutive integers. Thus, the parity of m+m^2-n depends on the parity of n only.
Thus, condition 2) is sufficient.
Therefore, B is the answer.
Answer: B
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.
m+m^2-n = m(m+1) - n. Now, m(m+1) is an even number since m(m+1) is the product of two consecutive integers. Thus, the parity of m+m^2-n depends on the parity of n only.
Thus, condition 2) is sufficient.
Therefore, B is the answer.
Answer: B
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