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Math Revolution GMAT Instructor V
Joined: 16 Aug 2015
Posts: 9252
GMAT 1: 760 Q51 V42
GPA: 3.82
Re: Overview of GMAT Math Question Types and Patterns on the GMAT  [#permalink]

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[Math Revolution GMAT math practice question]

What is the smallest integer n such that 28 is a factor of n!?

A. 8
B. 10
C. 12
D. 14
E. 16

=>

Since 2 = 2^1, 4 = 2^2, 6 = 2^1*3, 8 = 2^3, 10=2^1*5, the prime factorization of 10! = 1*2*3*…*10 has the form 28*m for some integer m, where m and 2 are relatively prime. Note that 9! = 2^7*k for some integer k, where k and 2 are relatively prime.

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GMAT 1: 760 Q51 V42
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[Math Revolution GMAT math practice question]

If m and n are positive integers, is 3^{4m+2}+n divisible by 5?

1) m=3
2) n=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.

The units digits of 3^k have period 4 as they form the cycle 3 -> 9 -> 7 -> 1.
3^{4m+2} has 9 as its units digit if 3^{4m+2} has units digit 9, regardless of the value of m.
Thus, the divisibility of 3^{4m+2}+n by 5 relies on the variable n only.

Therefore, the correct answer is B.
_________________
Math Revolution GMAT Instructor V
Joined: 16 Aug 2015
Posts: 9252
GMAT 1: 760 Q51 V42
GPA: 3.82
Re: Overview of GMAT Math Question Types and Patterns on the GMAT  [#permalink]

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[Math Revolution GMAT math practice question]

If x^2+y^2=28 and xy=11, what is the value of (x+y)^2?

A. 28
B. 39
C. 50
D. 61
E. 72

=>

(x+y)^2 = x^2 + 2xy + y^2 = x^2 + y^2 + 2xy = 28 + 2*11 = 28 + 22 = 50.

_________________
Math Revolution GMAT Instructor V
Joined: 16 Aug 2015
Posts: 9252
GMAT 1: 760 Q51 V42
GPA: 3.82
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[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

=>

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.

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: 9252
GMAT 1: 760 Q51 V42
GPA: 3.82
Re: Overview of GMAT Math Question Types and Patterns on the GMAT  [#permalink]

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[Math Revolution GMAT math practice question]

If x is integer and 3|x|+x<4, what is the value of x?

1) x<0
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.

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 original condition:
There are two cases to consider.
Case 1) x ≥ 0:
3|x|+x < 4
=> 3x + x < 4
=> 4x < 4
=> x < 1
=> 0 ≤ x < 1

Case 2) x < 0:
-3x+x < 4
=> -2x < 4
=> x > -2
=> -2 < x <0

Therefore, x is an integer with -2 < x < 1. Thus, the original condition tells us that x = -1 or 0.

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)
Since x < 0, we must have x = -1 as the original condition tells us that x = 0 or x = -1.
Condition 1) is sufficient, because it yields a unique solution.

Condition 2)
Both x = 0 and x = -1 satisfy condition 2).
Since it does not yield a unique solution, condition 2) is not sufficient.

_________________
Math Revolution GMAT Instructor V
Joined: 16 Aug 2015
Posts: 9252
GMAT 1: 760 Q51 V42
GPA: 3.82
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[Math Revolution GMAT math practice question]

In the xy-plane, a circle has center (0,0) and radius 5. Is the point (r,s) inside or on the circle?

1) -3 < r < 3
2) -4 < s < 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.

The inequality satisfied by points inside or on the circle is r^2+s^2≤5^2=25.

Since we have 2 variables (r and s) and 0 equations, C is most likely to be the answer. So, we should consider conditions 1) & 2) together first.

Conditions 1) & 2):
Since -3<r<3 and -4<s<4, we have 0≤r^2<3^2=9 and 0≤s^2<4^2=16. Thus, 0≤r^2+s^2<9+16=25 and both conditions together are sufficient.

Attachment: 1.17.png [ 18.37 KiB | Viewed 333 times ]

_________________
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Joined: 16 Aug 2015
Posts: 9252
GMAT 1: 760 Q51 V42
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[Math Revolution GMAT math practice question]

If x, y are integers, is x^2+x+y an odd integer?

1) x is an odd integer
2) y 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.

The first step of the VA (Variable Approach) method is to modify the original condition and the question. We then recheck the question.

The parity of x^2+x+y = x(x+1) + y is same as the parity of y, since x^2+x = x(x+1) is the product of two consecutive integers and so it is always an even integer.
Thus, asking whether x^2+x+y = x(x+1) + y is odd is equivalent to asking whether y is odd.

_________________
Math Revolution GMAT Instructor V
Joined: 16 Aug 2015
Posts: 9252
GMAT 1: 760 Q51 V42
GPA: 3.82
Re: Overview of GMAT Math Question Types and Patterns on the GMAT  [#permalink]

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[Math Revolution GMAT math practice question]

Three machines have equal constant work rates. It takes h + 3 hours to produce 360 toys when 2 machines work together, and it takes h hours to produce 360 toys when 3 machines work together. How many toys can each machine produce per hour when working on its own?

A. 12
B. 15
C. 18
D. 20
E. 24

=>

The work rate for each machine is given by 360 / {2(h+3)} = 180 / ( h + 3 ). Another expression for this work rate is 360 / (3h) = 120 / h.
Thus, 180 / ( h + 3 ) = 120 / h or 3 / ( h + 3 ) = 2 / h.
So, 3h = 2(h+3) and h = 6.
The sum of the work rates of the 3 machines is 360 / 6 = 60.
The work rate of each machine is 60/3 = 20 toys / hour.

_________________
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Joined: 16 Aug 2015
Posts: 9252
GMAT 1: 760 Q51 V42
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[Math Revolution GMAT math practice question]

If the median and the range of the data set 1, 1, 2, 2, 3, 3, 4, 4, m are the same, which of the following could be the value of m?

A. -1
B. 0
C. 1
D. 2
E. 3

=>

If 3 ≤ m ≤ 4, then the median is 3 and the range of the data is 3. Thus, any number between 3 and 4, inclusive, is possible.
If 2 ≤ m < 3, then the median is m and the range of the data is 3. Thus, there is no possible value of m in the interval 2 ≤ m < 3.
If m < 2, then the median is 2 and the range of the data is greater than 2. Thus, there is no possible value of m less than 2.
If m > 4, then the median is 3 and the range of the data is greater than 3. Thus, there is no possible value of m greater than 4.

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Joined: 16 Aug 2015
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Re: Overview of GMAT Math Question Types and Patterns on the GMAT  [#permalink]

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[Math Revolution GMAT math practice question]

If y=|x-1|+|x+1|, then y=?

1) x>-1
2) x<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.

There are three ranges of values of x to consider.
If x > 1, then y = | x – 1 | + | x + 1 | = x – 1 + x + 1 = 2x and we don’t have a unique value of y.
If -1 ≤ x ≤ 1, then y = | x – 1 | + | x + 1 | = - ( x – 1 ) + x + 1 = 2 and we have a unique value of y.
If x < 1, then y = | x – 1 | + | x + 1 | = -( x – 1 ) – ( x + 1 ) = -2x and we don’t have a unique value of y.

Asking for the value of y is equivalent asking if -1 ≤ x ≤ 1.
Both conditions yield the inequality -1 < x < 1, when applied together. Therefore, both conditions are sufficient, when taken together.

In inequality questions, the law “Question is King” tells us that if the solution set of the question includes the solution set of the condition, then the condition is sufficient.
_________________
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Joined: 16 Aug 2015
Posts: 9252
GMAT 1: 760 Q51 V42
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Re: Overview of GMAT Math Question Types and Patterns on the GMAT  [#permalink]

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[Math Revolution GMAT math practice question]

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.

_________________
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Joined: 16 Aug 2015
Posts: 9252
GMAT 1: 760 Q51 V42
GPA: 3.82
Re: Overview of GMAT Math Question Types and Patterns on the GMAT  [#permalink]

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[Math Revolution GMAT math practice question]

What is the measure of each interior angle of a regular decagon?

A. 72°
B. 108°
C. 120°
D. 135°
E. 144°

=>

The sum of all interior angles of n-gon is (n-2)*180°.
A decagon is a 10-gon.
The sum of all interior angles is (10-2)* 180° = 8*180°.
And each interior angle of a regular decagon has measure 8*180° / 10 = 8*18° = 144°.

_________________
Math Revolution GMAT Instructor V
Joined: 16 Aug 2015
Posts: 9252
GMAT 1: 760 Q51 V42
GPA: 3.82
Re: Overview of GMAT Math Question Types and Patterns on the GMAT  [#permalink]

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[GMAT math practice question]

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.

_________________
Math Revolution GMAT Instructor V
Joined: 16 Aug 2015
Posts: 9252
GMAT 1: 760 Q51 V42
GPA: 3.82
Re: Overview of GMAT Math Question Types and Patterns on the GMAT  [#permalink]

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[GMAT math practice question]

What is the value of (2^8 + 2^9 + 2^{10} + 2^{11}) / 32?

A. 80
B. 96
C. 100
D. 120
E. 160

=>

(2^8 + 2^9 + 2^{10} + 2^{11}) / 32
= (2^8 + 2^82^1 + 2^82^2 + 2^82^3) / 2^5
= 2^8(1 + 2^1 + 2^2 + 2^3) / 2^5
= 2^3(1+2+4+8)
= 8*15
= 120

_________________
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Joined: 16 Aug 2015
Posts: 9252
GMAT 1: 760 Q51 V42
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Re: Overview of GMAT Math Question Types and Patterns on the GMAT  [#permalink]

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[GMAT math practice question]

If a>b>c>d>0, is d<4?

1) 1/c + 1/d > 1/2
2) (1/a)+(1/b)+(1/c)+(1/d)=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.
The original condition a>b>c>d>0 is equivalent to 0 < 1/a < 1/b < 1/c < 1/d. The question asks if d < 4. This is equivalent to asking if 1/d > 1/4.

By condition 1, 1/d > 1/4 since 1/c < 1/d. So, d < 4. Condition 1) is sufficient.

Condition 2)
Since 0 < 1/a < 1/b < 1/c < 1/d and (1/a)+(1/b)+(1/c)+(1/d)=1, we have 1/a<1/d, 1/b<1/d, 1/c<1/d and 1/a +1/b + 1/c + 1/d < 1/d +1/d +1/d + 1/d = 4/d. Therefore, 1 < 4/d and d < 4.
Condition 2) is sufficient.

Note: This question is a CMT4(B) question: condition 1) is easy to work with and condition 2) is hard. For CMT4(B) questions, D is most likely to be the answer.
_________________
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Joined: 16 Aug 2015
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GMAT 1: 760 Q51 V42
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[GMAT math practice question]

What is the number of the x-intercepts of y=x^4-3x^3+2x^2?

A. one
B. two
C. three
D. four
E. five

=>

y = x^4-3x^3+2x^2
=> y = x^2(x^2-3x+2)
=> y = x^2(x-1)(z-2)
The x-intercepts occur when y = 0. This occurs when x = 0, x = 1 and x = 2.
There are three x-intercepts.

_________________
Math Revolution GMAT Instructor V
Joined: 16 Aug 2015
Posts: 9252
GMAT 1: 760 Q51 V42
GPA: 3.82
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[GMAT math practice question]

We define the harmonic mean of a set of numbers as the reciprocal of the average (arithmetic mean) of the reciprocals of the numbers. What is the harmonic mean of 20 and 30?

A. 22
B. 24
C. 25
D. 26
E. 28

=>

1 / { ( 1/20 + 1/30 ) / 2 } = 1 / { ( 3/60 + 2/60 ) / 2 } = 1 / { (5/60) / 2 } = 1 / { 5 / 120 } = 120 / 5 = 24.

_________________
Math Revolution GMAT Instructor V
Joined: 16 Aug 2015
Posts: 9252
GMAT 1: 760 Q51 V42
GPA: 3.82
Re: Overview of GMAT Math Question Types and Patterns on the GMAT  [#permalink]

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[GMAT math practice question]

Is 3(a-b)>0?

1) a^3>b^3
2) a>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.

Asking if 3(a-b)>0 is equivalent to asking if a – b > 0, or, equivalently, if a > b.
Thus, condition 2) is sufficient.

Condition 1)
a^3>b^3
=> a^3-b^3 > 0
=> (a-b)(a^2+ab+b^2) > 0
=> a - b > 0 since a^2+ab+b^2 > 0
=> a > b
Thus, condition 1) is sufficient too.

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.

_________________
Math Revolution GMAT Instructor V
Joined: 16 Aug 2015
Posts: 9252
GMAT 1: 760 Q51 V42
GPA: 3.82
Re: Overview of GMAT Math Question Types and Patterns on the GMAT  [#permalink]

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[GMAT math practice question]

Is |a|<1?

1) a^2<1
2) 1/(1-a^2)>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.

|a|<1
=> |a|^2 < 1
=> a^2 < 1
=> a^2-1 < 0
=> (a+1)(a-1) < 0
=> -1 < a < 1

Condition 1) is sufficient, since it is equivalent to the question.

Condition 2)
1/(1-a^2) > 0
=> (1-a^2) > 0
=> a^2-1 < 0
=> (a+1)(a-1) < 0
=> -1 < a < 1
Thus, condition 2) is sufficient, since it is also equivalent to the question.

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.
_________________
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Joined: 16 Aug 2015
Posts: 9252
GMAT 1: 760 Q51 V42
GPA: 3.82
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[GMAT math practice question]

Which of the following expressions is equal to 2^{32}-2^{31}-2^{30}?

A. 3*2^{29}
B. 5*2^{29}
C. 2^{30}
D. 3*2^{30}
E. 2^{31}

=>

2^{32}-2^{31}-2^{30}?
= (2^2)(2^{30})- (2^1)(2^{30})-2^{30}
= 4(2^{30})- 2(2^{30})-2^{30}
= (4-2-1)(2^{30})
=1(2^{30})=2^{30}

_________________ Re: Overview of GMAT Math Question Types and Patterns on the GMAT   [#permalink] 20 Feb 2019, 17:28

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# Overview of GMAT Math Question Types and Patterns on the GMAT  