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Math Revolution GMAT Instructor V
Joined: 16 Aug 2015
Posts: 8150
GMAT 1: 760 Q51 V42 GPA: 3.82

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MathRevolution wrote:
[GMAT math practice question]

(number properties) If $$n$$ is a positive integer, what is the value of $$n$$?

1) $$n(n-1)$$ is a prime number

2) $$n(n+1)$$ has $$4$$ factors

=>

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 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)
Only $$n =2$$ makes $$n(n-1)$$ a prime number.
Thus, condition 1) is sufficient.

Condition 2)
Integers with four factors have the form $$p*q$$ or $$p^3$$, where $$p$$ and $$q$$ are prime integers.
It is impossible to have $$n(n+1)=p^3$$, where $$n$$ is an integer and $$p$$ is a prime number.
The only time $$n(n+1) = pq$$ is when $$n(n+1) = 2*3$$ and $$n =2$$.
Condition 2) is sufficient since it yields a unique solution.

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.
_________________
Math Revolution GMAT Instructor V
Joined: 16 Aug 2015
Posts: 8150
GMAT 1: 760 Q51 V42 GPA: 3.82

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[GMAT math practice question]

(inequality) If $$x, y$$ and $$z$$ are integers with $$x<y<z$$, is $$z>4$$?

$$1) x+y+z=12$$

$$2) x<4$$
_________________
Math Revolution GMAT Instructor V
Joined: 16 Aug 2015
Posts: 8150
GMAT 1: 760 Q51 V42 GPA: 3.82

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MathRevolution wrote:
[GMAT math practice question]

(number properties) $$m$$ and $$n$$ are positive integers greater than $$1$$. Is $$m^n$$ a perfect square?

1) $$m$$ is an odd integer
2) $$n$$ is an odd 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 $$2$$ 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)
If $$m = 9$$ and $$n = 3$$, then $$mn = 9^3 = (3^2)^3 = 3^6 = (3^3)^2 = 27^2,$$ which is a perfect square, and the answer is ‘yes’.
If $$m = 3$$ and $$n = 3$$, then $$mn = 3^3 = (3)^3 = 27$$, which is not a perfect square, and the answer is ‘no’.
Both conditions together are not sufficient, since they don’t yield a unique answer.

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.
_________________
Math Revolution GMAT Instructor V
Joined: 16 Aug 2015
Posts: 8150
GMAT 1: 760 Q51 V42 GPA: 3.82

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[GMAT math practice question]

(number property) If $$m$$ and $$n$$ are positive integers, what is the remainder when $$12^{mn}$$ is divided by $$13$$?

1) m=even
2) n=odd
_________________
Math Revolution GMAT Instructor V
Joined: 16 Aug 2015
Posts: 8150
GMAT 1: 760 Q51 V42 GPA: 3.82

### Show Tags

MathRevolution wrote:
[GMAT math practice question]

(inequality) If $$x, y$$ and $$z$$ are integers with $$x<y<z$$, is $$z>4$$?

$$1) x+y+z=12$$

$$2) x<4$$

=>

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.

From condition 1), since $$x + y + z = 12$$, their average is $$4$$
.
The maximum of three numbers is greater than or equal to their average. Thus, $$z ≥ 4$$.

Indeed, we must have $$z > 4$$ since $$x, y$$ and $$z$$ are different for the following reasons.

If $$z = 4$$, then $$x + y = 8$$. But $$x < y < z (= 4)$$, so this is impossible.

If $$z ≤ 4$$, then $$x < y < z ≤ 4$$, and we must have $$x < 4$$ and $$y < 4$$.

This implies that $$x + y + z < 4 + 4 + 4 = 12$$ and $$x + y + z ≠ 12$$ , which contradicts condition 1).

Thus, $$z > 4$$.

Condition 1) is sufficient.

Condition 2)
If $$x = 3, y = 4$$ and $$z = 5$$, then $$z > 4$$ and the answer is ‘yes’.

If $$x = 1, y = 2$$ and $$z = 3$$, then $$z < 4$$ and the answer is ‘no’.

Condition 2) is not sufficient since it does not yield a unique answer.

_________________
Math Revolution GMAT Instructor V
Joined: 16 Aug 2015
Posts: 8150
GMAT 1: 760 Q51 V42 GPA: 3.82

### Show Tags

MathRevolution wrote:
[GMAT math practice question]

(number property) If $$m$$ and $$n$$ are positive integers, what is the remainder when $$12^{mn}$$ is divided by $$13$$?

1) m=even
2) n=odd

=>

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.
In remainder questions, you get the same answer if you do the divisions first and calculate with the remainders or do the calculations first and then find the remainder.
Since $$12 = 13*1 + (-1), 12$$, the remainder when $$12^{mn}$$ is divided by $$13$$ is the same as the remainder when $$(-1)^{mn}$$ is divided by $$13$$.

If $$mn$$ is an odd number, $$(-1)^{mn} = -1$$ and if $$mn$$ is an even number, $$(-1)^{mn} = 1$$.

The question asks if $$mn$$ is an odd number or an even number.

Condition 1)
If $$m$$ is an even integer, $$mn$$ is an even number and the remainder when $$12^{mn}$$ is divided by $$13$$ is $$1$$.

Condition 1) is sufficient, since it yields a unique answer.

Condition 2)
If $$m = 2$$ and $$n = 1$$, then $$12^{2*1} = 12^2 = 144 = 13*11 + 1$$ and the remainder is $$1$$.

If $$m = 1$$ and $$n = 1$$, then $$12^{1*1} = 12^1 = 12 = 13*0 + 12$$ and the remainder is $$12$$.

Condition 2) is not sufficient since it does not yield a unique answer.

_________________
Math Revolution GMAT Instructor V
Joined: 16 Aug 2015
Posts: 8150
GMAT 1: 760 Q51 V42 GPA: 3.82

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[GMAT math practice question]

(function) If $$f(x)=ax^2+bx+c$$, where $$a, b$$ and $$c$$ are integers, is $$b=0$$?

$$1) f(10)=f(-10)=0$$

$$2) f(0)f(10)=0$$
_________________
Math Revolution GMAT Instructor V
Joined: 16 Aug 2015
Posts: 8150
GMAT 1: 760 Q51 V42 GPA: 3.82

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[GMAT math practice question]

(geometry) What is the volume of a sphere?

1) The circumference of the sphere is $$6π$$

2) The surface area of the sphere is $$36π$$
_________________
Math Revolution GMAT Instructor V
Joined: 16 Aug 2015
Posts: 8150
GMAT 1: 760 Q51 V42 GPA: 3.82

### Show Tags

MathRevolution wrote:
[GMAT math practice question]

(function) If $$f(x)=ax^2+bx+c$$, where $$a, b$$ and $$c$$ are integers, is $$b=0$$?

$$1) f(10)=f(-10)=0$$

$$2) f(0)f(10)=0$$

=>

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.
If $$b = 0$$, then the graph of $$f(x)$$ is symmetric about the y-axis.

Condition 1) tells us that $$b = 0$$ because there are two possible ways that this graph can lie in the $$xy$$-plane.

Attachment: 4.29.png [ 9.75 KiB | Viewed 295 times ]

This implies that $$f(x) = a(x+10)(x-10)= a(x^2-100) = ax^2 – 100a$$ and the middle term of $$f(x)$$ is $$0$$.

Thus, condition 1) is sufficient.

Condition 2)
Condition 2) is equivalent to the statement that $$f(0) = 0$$ or $$f(10) = 0$$.

If $$f(x) = x^2 – 100$$, then $$f(10) = 0$$ and $$b = 0.$$

If $$f(x) = x^2 – 10x$$, then $$f(0) = 0$$ and $$b = 10.$$

Condition 2) is not sufficient since it does not yield a unique answer.

_________________
Math Revolution GMAT Instructor V
Joined: 16 Aug 2015
Posts: 8150
GMAT 1: 760 Q51 V42 GPA: 3.82

### Show Tags

[GMAT math practice question]

(function) In the $$x-y$$ plane, the graph of $$y=f(x)=ax^2+bx+c$$ passes through ($$-2,0$$) and ($$2,0$$). Is $$f(4)<0$$?

$$1) f(0)>0$$

$$2) f(-3)<0$$
_________________
Math Revolution GMAT Instructor V
Joined: 16 Aug 2015
Posts: 8150
GMAT 1: 760 Q51 V42 GPA: 3.82

### Show Tags

MathRevolution wrote:
[GMAT math practice question]

(geometry) What is the volume of a sphere?

1) The circumference of the sphere is $$6π$$

2) The surface area of the sphere is $$36π$$

=>

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 we have $$1$$ variable, the radius $$r$$ for the sphere, and $$0$$ equations, D is most likely to be the answer. So, we should consider each condition on its own first.

Condition 1)
Since the circumference of the sphere is $$2πr = 6π$$, we must have $$r = 3$$.

The volume of the sphere is $$(\frac{4}{3})πr^2 = (\frac{4}{3})π(3)^2 = 12π.$$

Condition 1) is sufficient since it yields a unique answer,

Condition 2)
Since the surface area of the sphere is $$4πr^2 = 36π$$, we must have $$r = 3$$.

The volume of the sphere is $$(\frac{4}{3})πr^2 = (\frac{4}{3})π3^2 = 12π.$$

Condition 2) is sufficient since it yields a unique answer,

Note: Tip 1) of the VA method states that D is most likely to be the answer if condition 1) gives the same information as condition 2).

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.
_________________
Math Revolution GMAT Instructor V
Joined: 16 Aug 2015
Posts: 8150
GMAT 1: 760 Q51 V42 GPA: 3.82

### Show Tags

[GMAT math practice question]

(inequality) $$a, b, c, d$$ and e are real numbers with $$a<b<c<d<e$$. Is $$abcde$$ negative?

$$1) abc < 0$$

$$2) cde < 0$$
_________________
Math Revolution GMAT Instructor V
Joined: 16 Aug 2015
Posts: 8150
GMAT 1: 760 Q51 V42 GPA: 3.82

### Show Tags

MathRevolution wrote:
[GMAT math practice question]

(function) In the $$x-y$$ plane, the graph of $$y=f(x)=ax^2+bx+c$$ passes through ($$-2,0$$) and ($$2,0$$). Is $$f(4)<0$$?

$$1) f(0)>0$$

$$2) f(-3)<0$$

=>

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.
Since $$f(2) = f(-2) = 0$$, there are two possible ways that this graph can lie in the $$xy$$-plane.

Attachment: 51.png [ 9.96 KiB | Viewed 273 times ]

The question asks if $$a < 0$$.

Condition 1) tells us that $$a < 0$$ since $$f(0) > 0$$. Thus, condition 1) it is sufficient.

Condition 2) also tells us that $$a < 0$$ since $$f(-3) < 0.$$ Thus, it is sufficient too.

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.
_________________
Math Revolution GMAT Instructor V
Joined: 16 Aug 2015
Posts: 8150
GMAT 1: 760 Q51 V42 GPA: 3.82

### Show Tags

[GMAT math practice question]

(number property) Is the integer $$n$$ even?

1) There is a sum of $$n$$ consecutive integers that is even.

2) $$[\frac{n}{2}]$$ is an even number, where $$[n]$$ is the greatest integer less than or equal to $$n$$.
_________________
Math Revolution GMAT Instructor V
Joined: 16 Aug 2015
Posts: 8150
GMAT 1: 760 Q51 V42 GPA: 3.82

### Show Tags

MathRevolution wrote:
[GMAT math practice question]

(inequality) $$a, b, c, d$$ and e are real numbers with $$a<b<c<d<e$$. Is $$abcde$$ negative?

$$1) abc < 0$$

$$2) cde < 0$$

=>

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.

Since we have $$5$$ variables and $$4$$ equations, D is most likely to be the answer. So, we should consider each condition on its own first. If a question is an inequality, then inequalities in the original condition can be counted as equations.

Condition 1)
If $$a = -1, b = 1, c = 2, d = 3, e = 4,$$ then $$abcde < 0$$, and the answer is ‘yes’

If $$a = -4, b = -3, c = -2, d = -1, e = 1,$$ then $$abcde > 0,$$ and the answer is ‘no’.

Condition 1) is not sufficient since it doesn’t yield a unique answer.

Condition 2)
There are two cases to consider:
i) $$0$$ lies between $$c$$ and $$d$$

ii) $$0$$ is greater than $$e.$$

If $$0$$ lies between $$c$$ and $$d$$, then $$abcde < 0$$ and the answer is ‘yes’.

If $$0$$ is greater than $$e$$, then $$abcde < 0$$ and the answer is ‘yes’.

Condition 2) is sufficient since it gives a unique answer.

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.
_________________
Math Revolution GMAT Instructor V
Joined: 16 Aug 2015
Posts: 8150
GMAT 1: 760 Q51 V42 GPA: 3.82

### Show Tags

MathRevolution wrote:
[GMAT math practice question]

(number property) Is the integer $$n$$ even?

1) There is a sum of $$n$$ consecutive integers that is even.

2) $$[\frac{n}{2}]$$ is an even number, where $$[n]$$ is the greatest integer less than or equal to $$n$$.

=>

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 the sum of $$2, 3, 4, 5$$ and $$6$$ is $$20$$, which is an even number, and $$n=5$$ is an odd number, the answer is sometimes ‘no’.
Since the sum of $$1, 2, 3$$, and $$4$$ is $$10$$, which is an even number, and $$n=4$$ is an even number, the answer is sometimes ‘yes’.
Condition 1) is not sufficient since it does not yield a unique answer.

Condition 2)
If $$n = 4$$, then $$[\frac{n}{2}] = 2$$ is even, and $$n$$ is even.

If $$n = 5$$, then $$[\frac{n}{2}] = 2$$ is also even, but $$n$$ is not even.
Condition 2) is not sufficient since it does not yield a unique answer.

Conditions 1) & 2)
Since the sum of $$2, 3, 4, 5$$ and $$6$$ is $$20$$, which is an even number, and $$n=5$$ is an odd number such that $$[\frac{n}{2}] = 2$$ is even, the answer is sometimes ‘no’.

Since the sum of $$1, 2, 3$$, and $$4$$ is $$10$$, which is an even number, and $$n=4$$ is an even number such that $$[\frac{n}{2}] = 2$$ is even, the answer is sometimes ‘yes’.

Conditions 1) & 2) together are not sufficient, since they do not yield a unique answer.

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.
_________________
Math Revolution GMAT Instructor V
Joined: 16 Aug 2015
Posts: 8150
GMAT 1: 760 Q51 V42 GPA: 3.82

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[GMAT math practice question]

(function) Is $$x + y < -1$$?

1) $$|x|>x$$ and $$|y|>y$$

2) $$x^2+y^2>1$$
_________________
Math Revolution GMAT Instructor V
Joined: 16 Aug 2015
Posts: 8150
GMAT 1: 760 Q51 V42 GPA: 3.82

### Show Tags

[GMAT math practice question]

(number properties) If $$x, y,$$ and $$z$$ are integers, is $$xyz$$ a multiple of $$6$$?

1) $$x+y+z$$ is a multiple of $$6$$

2) $$x, y$$, and $$z$$ are consecutive
_________________
Math Revolution GMAT Instructor V
Joined: 16 Aug 2015
Posts: 8150
GMAT 1: 760 Q51 V42 GPA: 3.82

### Show Tags

MathRevolution wrote:
[GMAT math practice question]

(function) Is $$x + y < -1$$?

1) $$|x|>x$$ and $$|y|>y$$

2) $$x^2+y^2>1$$

=>

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.

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)
Condition 1) tells us that $$x$$ and $$y$$ are negative numbers.
By condition 2), $$(x+y)^2 = x^2 + 2xy +y^2 > x^2 + y^2 > 1$$ and $$(x+y)^2 > 1$$ since $$x$$ and $$y$$ are negative and $$xy > 0.$$

Since $$x + y < 0,$$ we must have $$x + y < -1.$$
Conditions 1) & 2) are sufficient, when considered together.

_________________
Math Revolution GMAT Instructor V
Joined: 16 Aug 2015
Posts: 8150
GMAT 1: 760 Q51 V42 GPA: 3.82

### Show Tags

[GMAT math practice question]

(absolute value) Is $$|x-1|<|x-3|$$?

$$1) x<2$$

$$2) x>-2$$
_________________ Re: Math Revolution DS Expert - Ask Me Anything about GMAT DS   [#permalink] 08 May 2019, 02:45

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# Math Revolution DS Expert - Ask Me Anything about GMAT DS  