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Math Revolution GMAT Instructor
Joined: 16 Aug 2015
Posts: 8456
GMAT 1: 760 Q51 V42
GPA: 3.82

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25 Jan 2019, 02:29
[Math Revolution GMAT math practice question]

(exponents) $$m+n=?$$

$$1) (4^m)(2^n)=16$$
$$2) (2^{2m})(4^n)=64$$
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MathRevolution: Finish GMAT Quant Section with 10 minutes to spare
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"Only $79 for 1 month Online Course" "Free Resources-30 day online access & Diagnostic Test" "Unlimited Access to over 120 free video lessons - try it yourself" Math Revolution GMAT Instructor Joined: 16 Aug 2015 Posts: 8456 GMAT 1: 760 Q51 V42 GPA: 3.82 Re: Math Revolution DS Expert - Ask Me Anything about GMAT DS [#permalink] Show Tags 27 Jan 2019, 18:23 MathRevolution wrote: [Math Revolution GMAT math practice question] If $$x$$ and $$y$$ are non-zero numbers and $$x≠±y$$, then $$\frac{( x^2 + y^2 )}{( x^2 - y^2 )}=?$$ $$1) |\frac{x}{y}|=\frac{1}{3}$$ $$2) y=-3x$$ => 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. The question asks for the value of $$\frac{( x^2 + y^2 )}{( x^2 - y^2 )}= ( (\frac{x}{y})^2 + 1 ) / (\frac{x}{y})^2 – 1 ).$$ When a question asks for a ratio, if one condition includes a ratio and the other condition just gives a number, the condition including the ratio is most likely to be sufficient. Condition 1) Since $$|\frac{x}{y}|=\frac{1}{3}, \frac{x}{y} = ±(\frac{1}{3})$$, and $$\frac{( x^2 + y^2 )}{( x^2 - y^2 )}= ( (\frac{x}{y})^2 + 1 ) / ( (\frac{x}{y})^2 – 1 ) = ( (\frac{1}{3})^2 + 1 ) / ( (\frac{1}{3})^2 – 1) = (\frac{1}{9} + 1)/(\frac{1}{9}-1) = (\frac{10}{9})/(-\frac{8}{9}) = -\frac{10}{8} = -\frac{5}{4}.$$ Condition 1) is sufficient since it gives a unique solution. Condition 2) Since $$y = -3x, \frac{x}{y} = -\frac{1}{3}$$, and $$\frac{( x^2 + y^2 )}{( x^2 - y^2 )}= ( (\frac{x}{y})^2 + 1 ) / ( (\frac{x}{y})^2 – 1 ) = ( (-\frac{1}{3})^2 + 1 ) / ( (-\frac{1}{3})^2 – 1) = (\frac{1}{9} + 1)/(\frac{1}{9}-1) = (\frac{10}{9})/(-\frac{8}{9}) = -\frac{10}{8} = -\frac{5}{4}.$$ Condition 2) is sufficient since it gives a unique solution. Therefore, the answer is D. Answer: D 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. _________________ MathRevolution: Finish GMAT Quant Section with 10 minutes to spare The one-and-only World’s First Variable Approach for DS and IVY Approach for PS with ease, speed and accuracy. "Only$79 for 1 month Online Course"
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Math Revolution GMAT Instructor
Joined: 16 Aug 2015
Posts: 8456
GMAT 1: 760 Q51 V42
GPA: 3.82

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27 Jan 2019, 18:25
MathRevolution wrote:
[Math Revolution GMAT math practice question]

(exponents) $$m+n=?$$

$$1) (4^m)(2^n)=16$$
$$2) (2^{2m})(4^n)=64$$

=>

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.

Condition 2) is equivalent to $$m + n = 3$$ as shown below:
$$(2^{2m})(4^n)=64$$
$$=> (2^{2m})(2^{2n})=2^6$$
$$=> 2^{2m+2n}=2^6$$
$$=> 2m+2n = 6$$
$$=> m + n = 3$$
Condition 2) is sufficient.

Condition 1)
$$(4^m)(2^n)=16$$
$$=> (2^{2m})(2^n)=2^4$$
$$=> 2^{2m+n}=2^4$$
$$=> 2m+n = 4$$
If $$m = 1$$ and $$n =2$$, then $$m + n = 3$$.
If $$m = 0$$ and $$n = 4,$$ then $$m + n = 4.$$
Since it does not yield a unique solution, condition 1) is not sufficient.

_________________
MathRevolution: Finish GMAT Quant Section with 10 minutes to spare
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"Only $79 for 1 month Online Course" "Free Resources-30 day online access & Diagnostic Test" "Unlimited Access to over 120 free video lessons - try it yourself" Math Revolution GMAT Instructor Joined: 16 Aug 2015 Posts: 8456 GMAT 1: 760 Q51 V42 GPA: 3.82 Math Revolution DS Expert - Ask Me Anything about GMAT DS [#permalink] Show Tags 28 Jan 2019, 01:37 [GMAT math practice question] (number properties) If $$n$$ is positive integer, is $$4^n+n^2+1$$ divisible by $$2$$? 1) $$n$$ is a multiple of $$4$$ 2) $$n$$ is a multiple of $$6$$ _________________ MathRevolution: Finish GMAT Quant Section with 10 minutes to spare The one-and-only World’s First Variable Approach for DS and IVY Approach for PS with ease, speed and accuracy. "Only$79 for 1 month Online Course"
"Free Resources-30 day online access & Diagnostic Test"
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Math Revolution GMAT Instructor
Joined: 16 Aug 2015
Posts: 8456
GMAT 1: 760 Q51 V42
GPA: 3.82

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29 Jan 2019, 01:00
[GMAT math practice question]

(function) If operation $$#$$ represents one of addition, subtraction, multiplication, and division, what is the value of $$0#1$$?

$$1) 2#1 = 2$$
$$2) 4#2 = 2$$
_________________
MathRevolution: Finish GMAT Quant Section with 10 minutes to spare
The one-and-only World’s First Variable Approach for DS and IVY Approach for PS with ease, speed and accuracy.
"Only $79 for 1 month Online Course" "Free Resources-30 day online access & Diagnostic Test" "Unlimited Access to over 120 free video lessons - try it yourself" GMAT Club Legend Joined: 18 Aug 2017 Posts: 5734 Location: India Concentration: Sustainability, Marketing GPA: 4 WE: Marketing (Energy and Utilities) Math Revolution DS Expert - Ask Me Anything about GMAT DS [#permalink] Show Tags Updated on: 29 Jan 2019, 06:13 MathRevolution wrote: [GMAT math practice question] (function) If operation $$#$$ represents one of addition, subtraction, multiplication, and division, what is the value of $$0#1$$? $$1) 2#1 = 2$$ $$2) 4#2 = 2$$ #1 2#1=2 # can be multiplication or division so sufficient as doing both operations we would get answer as 0 for $$0#1$$ #2 4#2 =2 # can be subtraction of division; in sufficient IMO A Originally posted by Archit3110 on 29 Jan 2019, 03:07. Last edited by Archit3110 on 29 Jan 2019, 06:13, edited 1 time in total. GMAT Club Legend Joined: 18 Aug 2017 Posts: 5734 Location: India Concentration: Sustainability, Marketing GPA: 4 WE: Marketing (Energy and Utilities) Re: Math Revolution DS Expert - Ask Me Anything about GMAT DS [#permalink] Show Tags 29 Jan 2019, 03:16 MathRevolution wrote: [GMAT math practice question] (number properties) If $$n$$ is positive integer, is $$4^n+n^2+1$$ divisible by $$2$$? 1) $$n$$ is a multiple of $$4$$ 2) $$n$$ is a multiple of $$6$$ #1 n can be 0,4,8,12... and always the sum of $$4^n+n^2+1$$ would be odd which is not divisible by 2 ; sufficient #2 n can be 0,6,12,18... and always the sum of $$4^n+n^2+1$$ would be odd which is not divisible by 2 ; sufficient IMO D GMATH Teacher Status: GMATH founder Joined: 12 Oct 2010 Posts: 935 Re: Math Revolution DS Expert - Ask Me Anything about GMAT DS [#permalink] Show Tags 29 Jan 2019, 06:02 MathRevolution wrote: [GMAT math practice question] (function) If operation $$#$$ represents one of addition, subtraction, multiplication, and division, what is the value of $$0#1$$? $$1) 2#1 = 2$$ $$2) 4#2 = 2$$ $$?\,\, = \,\,0\# 1\,\,\,\,\,\,\left( {\# \in \left\{ { + , - , \times , \div } \right\}} \right)$$ $$\left( 1 \right)\,\,2\# 1 = 2\,\,\,\, \Rightarrow \left\{ \matrix{ \,\,\# = \times \,\,\,\,\,\, \Rightarrow \,\,\,\,\,\,? = 0 \hfill \cr \,\,{\rm{OR}} \hfill \cr \,\,\# = \div \,\,\,\,\,\, \Rightarrow \,\,\,\,\,\,? = 0 \hfill \cr} \right.\,\,\,\, \Rightarrow \,\,\,\,\,\,{\rm{SUFF}}{\rm{.}}$$ $$\left( 2 \right)\,\,\,\,4\# 2 = 2\,\,\,\, \Rightarrow \left\{ \matrix{ \,\,\# = - \,\,\,\,\,\, \Rightarrow \,\,\,\,\,\,? = - 1 \hfill \cr \,\,{\rm{OR}} \hfill \cr \,\,\# = \div \,\,\,\,\,\, \Rightarrow \,\,\,\,\,\,? = 0 \hfill \cr} \right.\,\,\,\, \Rightarrow \,\,\,\,\,\,{\rm{INSUFF}}{\rm{.}}$$ The correct answer is therefore (A). We follow the notations and rationale taught in the GMATH method. Regards, Fabio. _________________ Fabio Skilnik :: GMATH method creator (Math for the GMAT) Our high-level "quant" preparation starts here: https://gmath.net Math Revolution GMAT Instructor Joined: 16 Aug 2015 Posts: 8456 GMAT 1: 760 Q51 V42 GPA: 3.82 Re: Math Revolution DS Expert - Ask Me Anything about GMAT DS [#permalink] Show Tags 30 Jan 2019, 02:20 MathRevolution wrote: [GMAT math practice question] (number properties) If $$n$$ is positive integer, is $$4^n+n^2+1$$ divisible by $$2$$? 1) $$n$$ is a multiple of $$4$$ 2) $$n$$ is a multiple of $$6$$ => 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. Since $$4^n$$ is a multiple of $$2$$, we only need to look at $$n^2+1$$. If $$n$$ is an odd number, $$4^n+n^2+1$$ is divisible by $$2$$. If $$n$$ is an even number, $$4^n+n^2+1$$ is not divisible by $$2$$. The question asks if $$n$$ is an odd number. Thus, each of conditions is sufficient. Therefore, D is the answer. Answer: D _________________ MathRevolution: Finish GMAT Quant Section with 10 minutes to spare The one-and-only World’s First Variable Approach for DS and IVY Approach for PS with ease, speed and accuracy. "Only$79 for 1 month Online Course"
"Free Resources-30 day online access & Diagnostic Test"
"Unlimited Access to over 120 free video lessons - try it yourself"
Math Revolution GMAT Instructor
Joined: 16 Aug 2015
Posts: 8456
GMAT 1: 760 Q51 V42
GPA: 3.82

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30 Jan 2019, 02:22
[GMAT math practice question]

Five data values are $$11, 14, 16, 18$$ and $$x$$. What is the value of $$x$$?

1) The mode of the $$5$$ data values is $$11$$
2) The average (arithmetic mean) of the $$5$$ data values is $$14$$
_________________
MathRevolution: Finish GMAT Quant Section with 10 minutes to spare
The one-and-only World’s First Variable Approach for DS and IVY Approach for PS with ease, speed and accuracy.
"Only $79 for 1 month Online Course" "Free Resources-30 day online access & Diagnostic Test" "Unlimited Access to over 120 free video lessons - try it yourself" Math Revolution GMAT Instructor Joined: 16 Aug 2015 Posts: 8456 GMAT 1: 760 Q51 V42 GPA: 3.82 Math Revolution DS Expert - Ask Me Anything about GMAT DS [#permalink] Show Tags Updated on: 31 Jan 2019, 02:18 MathRevolution wrote: [GMAT math practice question] (function) If operation $$#$$ represents one of addition, subtraction, multiplication, and division, what is the value of $$0#1$$? $$1) 2#1 = 2$$ $$2) 4#2 = 2$$ => 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 operation is considered as a variable. Since we have $$1$$ variable and $$0$$ equations, D is most likely to be the answer. So, we should consider each condition on its own first. Condition 1) Since $$2#1 = 2$$, $$#$$ is one of the operations, multiplication and division. If $$#$$ is the multiplication operation, then $$0#1 = 0$$. If $$#$$ is the division operation, then $$0#1 = 0.$$ Since condition 1) yields a unique solution, it is sufficient. Condition 2) Since $$4#2 = 2, #$$ is one of the operations, subtraction and division. If $$#$$ is the subtraction operation, then $$0#1 = -1$$. If $$#$$ is the division operation, then $$0#1 = 0$$. Since condition 2) doesn’t yield a unique solution, it is not sufficient. Therefore, A is the answer. Answer: A _________________ MathRevolution: Finish GMAT Quant Section with 10 minutes to spare The one-and-only World’s First Variable Approach for DS and IVY Approach for PS with ease, speed and accuracy. "Only$79 for 1 month Online Course"
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Originally posted by MathRevolution on 31 Jan 2019, 02:12.
Last edited by MathRevolution on 31 Jan 2019, 02:18, edited 1 time in total.
Math Revolution GMAT Instructor
Joined: 16 Aug 2015
Posts: 8456
GMAT 1: 760 Q51 V42
GPA: 3.82

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31 Jan 2019, 02:13
[GMAT math practice question]

(number properties) Is $$3$$ a factor of $$x$$?

1) $$x-3$$ is divisible by $$6$$
2) $$x+3$$ is divisible by $$6$$
_________________
MathRevolution: Finish GMAT Quant Section with 10 minutes to spare
The one-and-only World’s First Variable Approach for DS and IVY Approach for PS with ease, speed and accuracy.
"Only $79 for 1 month Online Course" "Free Resources-30 day online access & Diagnostic Test" "Unlimited Access to over 120 free video lessons - try it yourself" GMAT Club Legend Joined: 18 Aug 2017 Posts: 5734 Location: India Concentration: Sustainability, Marketing GPA: 4 WE: Marketing (Energy and Utilities) Re: Math Revolution DS Expert - Ask Me Anything about GMAT DS [#permalink] Show Tags 31 Jan 2019, 02:59 MathRevolution wrote: [GMAT math practice question] Five data values are $$11, 14, 16, 18$$ and $$x$$. What is the value of $$x$$? 1) The mode of the $$5$$ data values is $$11$$ 2) The average (arithmetic mean) of the $$5$$ data values is $$14$$ #1 mode of set is 11 so x=11 mode most repeated no in a set sufficient #2 avg of set = 14 sum of digits = 59+x = 70 x=11 sufficient IMO D GMAT Club Legend Joined: 18 Aug 2017 Posts: 5734 Location: India Concentration: Sustainability, Marketing GPA: 4 WE: Marketing (Energy and Utilities) Re: Math Revolution DS Expert - Ask Me Anything about GMAT DS [#permalink] Show Tags 31 Jan 2019, 03:02 MathRevolution wrote: [GMAT math practice question] (number properties) Is $$3$$ a factor of $$x$$? 1) $$x-3$$ is divisible by $$6$$ 2) $$x+3$$ is divisible by $$6$$ #1 sufficient x= 9,15,21 x-3 divisible by 6 and x a factor of 3 #2 sufficient x+3 divisible by 6 x=3,9,15,18 sufficient IMO D Math Revolution GMAT Instructor Joined: 16 Aug 2015 Posts: 8456 GMAT 1: 760 Q51 V42 GPA: 3.82 Re: Math Revolution DS Expert - Ask Me Anything about GMAT DS [#permalink] Show Tags 01 Feb 2019, 02:48 MathRevolution wrote: [GMAT math practice question] Five data values are $$11, 14, 16, 18$$ and $$x$$. What is the value of $$x$$? 1) The mode of the $$5$$ data values is $$11$$ 2) The average (arithmetic mean) of the $$5$$ data values is $$14$$ => 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 ($$x$$) and $$0$$ equations, D is most likely to be the answer. So, we should consider each condition on its own first. Condition 1) The data values include four different values and $$x$$. Since the mode of the five data values is $$11, x$$ must equal $$11$$. Condition 1) is sufficient. Condition 2) Calculating the mean of the five data values yields $$\frac{( 11 + 14 + 16 + 18 + x )}{5} = 14.$$ Solving for $$x$$ gives $$11 + 14 + 16 + 18 + x = 70$$ $$59 + x = 70$$ $$x = 11$$ Condition 2) is also sufficient. Therefore, D is the answer. Answer: D _________________ MathRevolution: Finish GMAT Quant Section with 10 minutes to spare The one-and-only World’s First Variable Approach for DS and IVY Approach for PS with ease, speed and accuracy. "Only$79 for 1 month Online Course"
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"Unlimited Access to over 120 free video lessons - try it yourself"
Math Revolution GMAT Instructor
Joined: 16 Aug 2015
Posts: 8456
GMAT 1: 760 Q51 V42
GPA: 3.82

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01 Feb 2019, 02:49
[GMAT math practice question]

(exponent) If $$p, x,$$ and $$y$$ are integers, $$\frac{x^p}{x^q}$$=?

$$1) p=q+4$$
$$2) x^q=16$$
_________________
MathRevolution: Finish GMAT Quant Section with 10 minutes to spare
The one-and-only World’s First Variable Approach for DS and IVY Approach for PS with ease, speed and accuracy.
"Only $79 for 1 month Online Course" "Free Resources-30 day online access & Diagnostic Test" "Unlimited Access to over 120 free video lessons - try it yourself" GMATH Teacher Status: GMATH founder Joined: 12 Oct 2010 Posts: 935 Re: Math Revolution DS Expert - Ask Me Anything about GMAT DS [#permalink] Show Tags 01 Feb 2019, 06:08 MathRevolution wrote: [GMAT math practice question] (exponent) If $$p, x,$$ and $$q$$ are integers, $$\frac{x^p}{x^q}$$=? $$1) p=q+4$$ $$2) x^q=16$$ $$p,x,q\,\,{\rm{ints}}$$ $$?\,\, = \,\,{{{x^p}} \over {{x^q}}}$$ $$\left( {1 + 2} \right)\,\,\,\left\{ \matrix{ \,p = q + 4 \hfill \cr \,{x^q} = 16 \hfill \cr} \right.\,\,\,\,\,\,::\,\,\,\,\,\,\left\{ \matrix{ \,{\rm{Take}}\,\,\left( {x,q,p} \right) = \left( {16,1,5} \right)\,\,\,\, \Rightarrow \,\,\,? = {16^4} \hfill \cr \,{\rm{Take}}\,\,\left( {x,q,p} \right) = \left( {2,4,8} \right)\,\,\,\, \Rightarrow \,\,\,? = {2^4} \hfill \cr} \right.$$ The correct answer is therefore (E). We follow the notations and rationale taught in the GMATH method. Regards, Fabio. _________________ Fabio Skilnik :: GMATH method creator (Math for the GMAT) Our high-level "quant" preparation starts here: https://gmath.net Math Revolution GMAT Instructor Joined: 16 Aug 2015 Posts: 8456 GMAT 1: 760 Q51 V42 GPA: 3.82 Re: Math Revolution DS Expert - Ask Me Anything about GMAT DS [#permalink] Show Tags 04 Feb 2019, 05:30 MathRevolution wrote: [GMAT math practice question] (number properties) Is $$3$$ a factor of $$x$$? 1) $$x-3$$ is divisible by $$6$$ 2) $$x+3$$ is divisible by $$6$$ => 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 (x) and 0 equations, D is most likely to be the answer. So, we should consider each condition on its own first. To solve remainder questions, plugging in numbers is recommended. Condition 1) If we plug in x = 9, then x – 3 = 6 is divisible by 6 and x is a multiple of 3. Condition 1) is sufficient. Condition 2) If we plug in x = 9, then x + 3 = 12 is divisible by 6 and x is a multiple of 3. Condition 2) is sufficient. Therefore, the answer is D. Answer: D _________________ MathRevolution: Finish GMAT Quant Section with 10 minutes to spare The one-and-only World’s First Variable Approach for DS and IVY Approach for PS with ease, speed and accuracy. "Only$79 for 1 month Online Course"
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Math Revolution GMAT Instructor
Joined: 16 Aug 2015
Posts: 8456
GMAT 1: 760 Q51 V42
GPA: 3.82

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04 Feb 2019, 05:36
MathRevolution wrote:
[GMAT math practice question]

(exponent) If $$p, x,$$ and $$y$$ are integers, $$\frac{x^p}{x^q}$$=?

$$1) p=q+4$$
$$2) x^q=16$$

=>

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 3 variables (x, y and z) and 0 equations, E 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)
If $$x = 2, p = 8$$ and $$q = 4$$, then $$\frac{x^p}{x^q} = x^{p-q} = x^4 = 2^4 = 16.$$
If $$x = 16, p = 5$$ and $$q = 1$$, then $$\frac{x^p}{x^q} = x^{p-q} = x^4 = 16^4 = 2^{16} = 65536.$$
Since they do not yield a unique solution, both conditions are not sufficient, when considered together.

Note: This question is related to finding a hidden 1.
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.
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"Only $79 for 1 month Online Course" "Free Resources-30 day online access & Diagnostic Test" "Unlimited Access to over 120 free video lessons - try it yourself" Math Revolution GMAT Instructor Joined: 16 Aug 2015 Posts: 8456 GMAT 1: 760 Q51 V42 GPA: 3.82 Re: Math Revolution DS Expert - Ask Me Anything about GMAT DS [#permalink] Show Tags 04 Feb 2019, 05:38 [GMAT math practice question] (number properties) If $$x, y$$ are integers, is $$(x-y)(x+y)(x^2+y^2)$$ an odd number? 1) $$x$$ is an odd number 2) $$x-y$$ is an odd number _________________ MathRevolution: Finish GMAT Quant Section with 10 minutes to spare The one-and-only World’s First Variable Approach for DS and IVY Approach for PS with ease, speed and accuracy. "Only$79 for 1 month Online Course"
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Re: Math Revolution DS Expert - Ask Me Anything about GMAT DS   [#permalink] 04 Feb 2019, 05:38

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