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Math Revolution DS Expert - Ask Me Anything about GMAT DS

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Joined: 16 Aug 2015
Posts: 8563
GMAT 1: 760 Q51 V42
GPA: 3.82
Re: Math Revolution DS Expert - Ask Me Anything about GMAT DS  [#permalink]

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New post 13 Dec 2018, 23:54
MathRevolution wrote:
[Math Revolution GMAT math practice question]

(number property) If \(n\) is a positive integer, is \(\sqrt{n+1}\) an integer?

1) \(n\) is a multiple of \(8\)
2) \(n\) is the product of \(2\) consecutive even numbers


=>

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.

Condition 1)
If \(n = 8\), then \(\sqrt{n+1} = \sqrt{8+1} = \sqrt{9} = 3\) and the answer is ‘yes’.
If \(n = 16\), then \(\sqrt{n+1} = \sqrt{16+1} = \sqrt{17}\) and the answer is ‘no’.
Thus, condition 1) is not sufficient since it does not yield a unique solution.

Condition 2)
If \(n\) is the product of two consecutive even integers, then
\(n = 2k(2k+2) = 4k^2 + 4k\) for some integer \(k\).
\(\sqrt{n+1} = \sqrt{4k^2+4k+1} = \sqrt{(2k+1)^2} = 2k+1,\) and the answer is ‘yes’.
Thus, condition 2) is sufficient.

Therefore, B is the answer.
Answer: B

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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New post 13 Dec 2018, 23:55
[Math Revolution GMAT math practice question]

(number property) When \(n\) is a positive integer, is \(\frac{n}{4}\) an integer?

1) \(n - 1\) is not divisible by \(2\)
2) \(n + 1\) is not divisible by \(2\)
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Re: Math Revolution DS Expert - Ask Me Anything about GMAT DS  [#permalink]

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New post 16 Dec 2018, 17:36
MathRevolution wrote:
[Math Revolution GMAT math practice question]

(number properties) If \(m\) and \(n\) are positive integers, is \(mn\) an even number?

1) \(\frac{m}{n}\) is an even number.
2) \(m + n\) is an even number.


=>

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.

Modifying the question:
\(mn\) is an even number precisely when at least one of \(m\) and \(n\) is even. So,
the question asks if either \(m\) or \(n\) is even.

Condition 1:
If \(\frac{m}{n} = 2k\) for some integer \(k\), then \(m = 2kn\), which is an even number.
Thus, condition 1) is sufficient.

Condition 2)
If \(m = 2\) and \(n = 4\), then \(m + n = 6\) is even, and \(mn = 8\) is an even number, so the answer is ‘yes’.
If \(m = 1\) and \(n = 3\), then \(m + n = 4\) is even, and \(mn = 3\) is not an even number, so the answer is ‘no’.
Since it does not give us a unique answer, condition 2) is not sufficient.

Therefore, the correct answer is A.
Answer: A
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Re: Math Revolution DS Expert - Ask Me Anything about GMAT DS  [#permalink]

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New post 16 Dec 2018, 17:39
MathRevolution wrote:
[Math Revolution GMAT math practice question]

(number property) When \(n\) is a positive integer, is \(\frac{n}{4}\) an integer?

1) \(n - 1\) is not divisible by \(2\)
2) \(n + 1\) is not divisible by \(2\)


=>

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.

Modifying the question:
Asking whether \(\frac{n}{4}\) is an integer is equivalent to asking whether \(m\) is a multiple of \(4\).

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)
Since \(n – 1\) is not divisible by \(2, n – 1\) is an odd number and \(n\) is an even number.
If \(n = 4\), then \(n\) is a multiple of \(4\) and the answer is ‘yes’.
If \(n = 2,\) then \(n\) is not a multiple of \(4\) and the answer is ‘no’.
Thus, condition 1) is not sufficient, since it does not yield a unique solution.

Condition 2)
Since \(n + 1\) is not divisible by \(2, n + 1\) is an odd number and \(n\) is an even number.
If \(n = 4\), then \(n\) is a multiple of \(4\) and the answer is ‘yes’.
If \(n = 2\), then \(n\) is not a multiple of \(4\) and the answer is ‘no’.
Thus, condition 2) is not sufficient, since it does not yield a unique solution.

Conditions 1) & 2):
If \(n = 4,\) then neither \(n – 1\) nor \(n + 1\) is divisible by \(2\), but \(n\) is a multiple of \(4\) and the answer is ‘yes’.
If \(n = 2\), then neither \(n – 1\) nor \(n + 1\) is divisible by \(2\), and \(n\) is not a multiple of \(4\) and the answer is ‘no’.
Thus, both conditions together are not sufficient, since they do not yield a unique solution.

Therefore, the answer is E.
Answer: E

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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New post 16 Dec 2018, 23:39
[Math Revolution GMAT math practice question]

(number property) \([x]\) is the greatest integer less than or equal to \(x\). \(<x>\) is the least integer greater than or equal to \(x\). What is the value of \(x\)?

\(1) [x] = 2\)
\(2) <x> = 2\)
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New post 18 Dec 2018, 00:37
[Math Revolution GMAT math practice question]

(number properties) \(x\) and \(y\) are positive integers. Is \(y\) an even integer?

\(1) x^2+x=y+2\)
\(2) x = 2\)
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New post 19 Dec 2018, 04:59
MathRevolution wrote:
[Math Revolution GMAT math practice question]

(number property) \([x]\) is the greatest integer less than or equal to \(x\). \(<x>\) is the least integer greater than or equal to \(x\). What is the value of \(x\)?

\(1) [x] = 2\)
\(2) <x> = 2\)


=>

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.

\([x]\) is analyzed as follows.
If \(n ≤ x < n + 1\) for some integer \(n\), then \([x] = n.\)
\(<x>\) is analyzed as follows.
If \(n – 1 < x ≤ n\) for some integer \(n\), then \(<x> = n.\)

Since we have \(1\) variable (\(x\)) and \(0\) equations, D is most likely to be the answer. So, we should consider each condition on its own first.

Condition 1)
\([x] = 2\)
\(=> 2 ≤ x < 3\)
Thus, condition 1) is not sufficient, since it does not yield a unique solution.

Condition 2)
\(<x> = 2\)
\(=> 1 < x ≤ 2\)
Thus, condition 2) is not sufficient, since it does not yield a unique solution.

Conditions 1) & 2)
Only \(x = 2\) satisfies both conditions.
Since the answer is unique, both conditions together are sufficient.

Therefore, C is the answer.
Answer: C

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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New post 19 Dec 2018, 04:59
[Math Revolution GMAT math practice question]

(inequality) Is \(\frac{x}{y}>1\)?

\(1) x>y\)
\(2) x-y>1\)
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Re: Math Revolution DS Expert - Ask Me Anything about GMAT DS  [#permalink]

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New post 20 Dec 2018, 00:58
MathRevolution wrote:
[Math Revolution GMAT math practice question]

(number properties) \(x\) and \(y\) are positive integers. Is \(y\) an even integer?

\(1) x^2+x=y+2\)
\(2) x = 2\)


=>

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 \(2\) variables (\(x\) and \(y\)) and \(0\) equations, C is most likely to be the answer. So, we should consider conditions 1) & 2) together first. After comparing the number of variables and the number of equations, we can save time by considering conditions 1) & 2) together first.

Conditions 1) & 2):
Since \(y = x^2 + x – 2\) and \(x = 2\), we have \(y = 4\).
Since this answer is unique, both conditions together are sufficient.

Since this question is an integer question (one of the key question areas), CMT (Common Mistake Type) 4(A) of the VA (Variable Approach) method tells us that we should also check answers A and B.

Condition 1)
Since \(y = x^2 + x – 2 = (x-1)(x+2)\) and \(x\) is an integer, one of \(x – 1\) and \(x + 2\) is an even integer.
Thus, \(y\) is always an even integer and condition 1) is sufficient.

Condition 2)
Since it provides no information about y, condition 2) is not sufficient.


Therefore, A is the answer.
Answer: A

Normally, in problems which require 2 equations, such as those in which the original conditions include 2 variables, or 3 variables and 1 equation, or 4 variables and 2 equations, each of conditions 1) and 2) provide an additional equation. In these problems, the two key possibilities are that C is the answer (with probability 70%), and E is the answer (with probability 25%). Thus, there is only a 5% chance that A, B or D is the answer. This occurs in common mistake types 3 and 4. Since C (both conditions together are sufficient) is the most likely answer, we save time by first checking whether conditions 1) and 2) are sufficient, when taken together. Obviously, there may be cases in which the answer is A, B, D or E, but if conditions 1) and 2) are NOT sufficient when taken together, the answer must be E.
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New post 20 Dec 2018, 00:59
[Math Revolution GMAT math practice question]

(absolute value) Is \(|m-n|=|m|-|n|\) ?

\(1) m-n = 0\)
\(2) n = 0\)
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New post 20 Dec 2018, 23:38
MathRevolution wrote:
[Math Revolution GMAT math practice question]

(inequality) Is \(\frac{x}{y}>1\)?

\(1) x>y\)
\(2) x-y>1\)


=>

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.

\(\frac{x}{y}>1\)
\(=> xy>y^2\)
\(=> xy-y^2>0\)
\(=> y(x-y)>0\)
By condition 2), \(x-y > 1 > 0\), but we can’t determine whether \(y\) is positive from condition 1).

Therefore, E is the answer.
Answer: E
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New post 20 Dec 2018, 23:39
[Math Revolution GMAT math practice question]

(integer) \(n\) is a positive integer. Is \(\frac{n(n+1)(n+2)}{4}\) an even integer?

1) \(n\) is an even integer
2) \(1238 ≤ n ≤ 1240\)
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Re: Math Revolution DS Expert - Ask Me Anything about GMAT DS  [#permalink]

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New post 23 Dec 2018, 10:01
MathRevolution wrote:
MathRevolution wrote:
[Math Revolution GMAT math practice question]

(number property) If \(n\) is an integer greater than \(1\), what is the value of \(n\)?

1) \(n\) is a prime number
2) \(\frac{(n+2)}{n}\) is an integer


=>

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)
Since there are many prime numbers, condition 1) is not sufficient.

Condition 2)
If \(n = 1\), then \(\frac{(n+2)}{n} = 3\) is an integer.
If \(n = 2,\) then \(\frac{(n+2)}{n} = 2\) is an integer.
Since we don’t have a unique solution, condition 2) is not sufficient.

Conditions 1) & 2)
If \(n = 2\), then \(\frac{(n+2)}{n} = 2\) is an integer.
If \(n = 3\), then \(\frac{(n+2)}{n} = \frac{5}{2}\) is not integer.
If \(n\) is a prime number bigger than \(2\), \(\frac{(n+2)}{n}\) is not an integer.
Thus \(n = 2\) is the unique solution and both conditions together are sufficient.

Therefore, C is the answer.
Answer: C

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.

Please check the OA, Highlighted text is wrong. Stem alreay mention n as > 1.

Regards
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Re: Math Revolution DS Expert - Ask Me Anything about GMAT DS  [#permalink]

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New post 23 Dec 2018, 18:18
MathRevolution wrote:
[Math Revolution GMAT math practice question]

(absolute value) Is \(|m-n|=|m|-|n|\) ?

\(1) m-n = 0\)
\(2) n = 0\)


=>

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.

Squaring both sides of \(|m – n| = |m| - |n|\) yields
\(|m-n|^2=(|m|-|n|)^2\)
\(=> (m-n)^2=(|m|-|n|)^2\)
\(=> m^2+n^2-2mn=|m|^2+|n|^2-2|mn|\)
\(=> m^2+n^2-2mn=m^2+n^2-2|mn|\)
\(=> -2mn=-2|mn|\)
\(=> mn=|mn|\)
\(=> mn ≥ 0\)

Condition 1):
\(m – n = 0\) implies that \(m = n.\) So \(mn = n^2 ≥ 0\) for all values of \(n\).
Condition 1) is sufficient.

Condition 2):
If \(n = 0\), then \(0 = mn ≥ 0.\)
Condition 2) is sufficient.
Therefore, the correct answer is D.
Answer: D
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Re: Math Revolution DS Expert - Ask Me Anything about GMAT DS  [#permalink]

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New post 23 Dec 2018, 18:20
MathRevolution wrote:
[Math Revolution GMAT math practice question]

(integer) \(n\) is a positive integer. Is \(\frac{n(n+1)(n+2)}{4}\) an even integer?

1) \(n\) is an even integer
2) \(1238 ≤ n ≤ 1240\)


=>

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.

Asking for \(\frac{n(n+1)(n+2)}{4}\) to be an even integer is equivalent to asking for \(n(n+1)(n+2)\) to be a multiple of \(8\). If \(n\) is an even integer, \(n\) and \(n+2\) are consecutive even integers and a product of two consecutive even integers is a multiple of \(8\). Thus, condition 1) is sufficient.

Condition 2)
If \(n = 1238, n(n+1)(n+2)=1238*1239*1240\) is a multiple of \(8\) since \(1240\) is a multiple of \(8\).
If \(n = 1239, n(n+1)(n+2)=1239*1240*1241\) is a multiple of \(8\) since \(1240\) is a multiple of \(8\).
If \(n = 1240, n(n+1)(n+2)=1240*1241*1242\) is a multiple of \(8\) since \(1240\) is a multiple of \(8\).
Thus, condition 2) is sufficient.

Therefore, the answer is D.
Answer: D

Note: This question is a CMT4(B) question. Condition 1) is easy to understand and condition 2) is hard. When one condition is easy to understand, and the other is hard, D is most likely to be the answer.
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New post 24 Dec 2018, 00:01
[Math Revolution GMAT math practice question]

(number property) \(f(x)\) is the greatest prime factor of \(x\). If \(n\) is a positive integer less than \(10\), what is the value of \(n\)?

\(1) f(n) = f(1000)\)
\(2) f(n) = 5\)
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New post 24 Dec 2018, 10:17
ShankSouljaBoi wrote:
MathRevolution wrote:
MathRevolution wrote:
[Math Revolution GMAT math practice question]

(number property) If \(n\) is an integer greater than \(1\), what is the value of \(n\)?

1) \(n\) is a prime number
2) \(\frac{(n+2)}{n}\) is an integer


=>

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)
Since there are many prime numbers, condition 1) is not sufficient.

Condition 2)
If \(n = 1\), then \(\frac{(n+2)}{n} = 3\) is an integer.
If \(n = 2,\) then \(\frac{(n+2)}{n} = 2\) is an integer.
Since we don’t have a unique solution, condition 2) is not sufficient.

Conditions 1) & 2)
If \(n = 2\), then \(\frac{(n+2)}{n} = 2\) is an integer.
If \(n = 3\), then \(\frac{(n+2)}{n} = \frac{5}{2}\) is not integer.
If \(n\) is a prime number bigger than \(2\), \(\frac{(n+2)}{n}\) is not an integer.
Thus \(n = 2\) is the unique solution and both conditions together are sufficient.

Therefore, C is the answer.
Answer: C

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.

Please check the OA, Highlighted text is wrong. Stem alreay mention n as > 1.

Regards


You are right.
The question should be changed to the followings.

If \(n\) is a positive integer, what is the value of \(n\)?

Thank you for your comments.
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Re: Math Revolution DS Expert - Ask Me Anything about GMAT DS  [#permalink]

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New post 26 Dec 2018, 00:42
MathRevolution wrote:
[Math Revolution GMAT math practice question]

(number property) \(f(x)\) is the greatest prime factor of \(x\). If \(n\) is a positive integer less than \(10\), what is the value of \(n\)?

\(1) f(n) = f(1000)\)
\(2) f(n) = 5\)


=>

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)
Since \(1000 = 2^3*5^3, f(1000) = 5.\)
So, \(f(n) = 5\) and \(n = 5.\)
Thus, condition 1) is sufficient, since it gives a unique solution.

Condition 2)
Condition 2) is the same as condition 1).
Thus, condition 2) is also sufficient.

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.

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.
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Re: Math Revolution DS Expert - Ask Me Anything about GMAT DS  [#permalink]

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New post 26 Dec 2018, 00:43
[Math Revolution GMAT math practice question]

(number properties) If \(m\) and \(n\) are positive integers, is \(m + n\) an odd number?

1) \(\frac{m}{n}\) is an even number
2) \(m\) or \(n\) is an even number
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Re: Math Revolution DS Expert - Ask Me Anything about GMAT DS  [#permalink]

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New post 26 Dec 2018, 06:19
MathRevolution wrote:
[Math Revolution GMAT math practice question]

(number properties) If \(m\) and \(n\) are positive integers, is \(m + n\) an odd number?

1) \(\frac{m}{n}\) is an even number
2) \(m\) or \(n\) is an even number

\(m,n\,\,\, \geqslant 1\,\,\,{\text{ints}}\,\,\,\,\left( * \right)\)

\(m + n\,\,\,\,\mathop = \limits^? \,\,{\text{odd}}\,\,\,\,\,\mathop \Leftrightarrow \limits^{\left( * \right)} \,\,\,\,\boxed{\,\,\,?\,\,\,:\,\,\,\left( {m\,\,{\text{odd}}\,,\,\,n\,\,{\text{even}}} \right)\,\,\,{\text{or}}\,\,\,{\text{vice - versa}\,\,}\,\,}\)


\(\left( 1 \right)\,\,\,\frac{m}{n} = {\text{even}}\,\,\,\,\left\{ \begin{gathered}
\,{\text{Take}}\,\,\left( {m,n} \right) = \left( {2,1} \right)\,\,\,\, \Rightarrow \,\,\,\left\langle {{\text{YES}}} \right\rangle \,\, \hfill \\
\,{\text{Take}}\,\,\left( {m,n} \right) = \left( {4,2} \right)\,\,\,\, \Rightarrow \,\,\,\left\langle {{\text{NO}}} \right\rangle \,\, \hfill \\
\end{gathered} \right.\)

\(\left( 2 \right)\,\,\,m\,\,{\text{even}}\,\,\,{\text{or}}\,\,\,n\,\,{\text{even}}\,\,\,\,\left\{ \begin{gathered}
\,\left( {\operatorname{Re} } \right){\text{Take}}\,\,\left( {m,n} \right) = \left( {2,1} \right)\,\,\,\, \Rightarrow \,\,\,\left\langle {{\text{YES}}} \right\rangle \,\, \hfill \\
\,\left( {\operatorname{Re} } \right){\text{Take}}\,\,\left( {m,n} \right) = \left( {4,2} \right)\,\,\,\, \Rightarrow \,\,\,\left\langle {{\text{NO}}} \right\rangle \,\, \hfill \\
\end{gathered} \right.\,\,\,\,\,\,\,\,\,\,\,\,\,\, \Rightarrow \,\,\,\,\,\,\,\,\,\,\,\,\,\left( {\text{E}} \right)\)


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

Regards,
Fabio.

P.S.: "A or B" means "only A", "only B" or BOTH.
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Re: Math Revolution DS Expert - Ask Me Anything about GMAT DS   [#permalink] 26 Dec 2018, 06:19

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