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Math Revolution GMAT Instructor V
Joined: 16 Aug 2015
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[GMAT math practice question]

In the x-y plane, y = f(x) = ax^2+bx+c passes through (1,0) and (2,0). Is f(3)>0?

1) a > 0
2) f(0) > 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(x) passes through (1,0) and (2,0), 1 and 2 are roots of f(x) and we have f(x) = a(x-1)(x-2). If f(x) is concave up, then f(3) > 0. Thus, knowing that f(x) is concave up will allow us to answer the question. In addition, since we have 3 variables (a, b, c) and 2 conditions, D is most likely to be the answer.

Condition 1)
If a > 0, then f(x) is concave up.
Thus, condition 1) is sufficient.

Condition 2)
Since 1 and 2 are roots of f(x) = 0, f(0) > 0 implies that f(x) is concave up.
Thus, condition 2) is sufficient, too.

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).
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[GMAT math practice question]

In the xy-plane, line L has equation y=mx+b, where 1<m<5. If line L passes through (1,1), which of the following points also lies on line L?

A. (2, -1)
B. (2,0)
C. (2,1)
D. (2,2)
E. (2,3)

=>

When we plug in the values x = 1 and y = 1, we obtain 1 = m + b or b = 1 – m.
All answer choices have x-coordinates of 2. Plugging 2 in for the variable x yields
y = 2m + b = 2m + 1 – m = m + 1.
Since 1 < m < 5, we must have 2 < m + 1 < 6 and 2 < y < 6.
Thus, (2,3) is the only point that can lie on the line.

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[GMAT math practice question]

|m-n|=?

1) m and n are integers
2) mn=13

=>

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 (m and n) 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 m and n are integers, the pairs of solutions of the equation mn=13 are
m = 1, n = 13; m = 13, n = 1; m = -1, n = -13; and m = -13, n = -1.
If m = 1, n = 13, then | m – n | = | 1 – 13 | = 12.
If m = 13, n = 1, then | m – n | = | 13 – 1 | = 12.
If m = -1, n = -13, then | m – n | = | -1 – (-13) | = 12.
If m = -13, n = -1, then | m – n | = | -13 – (-1) | = 12.

Since this question is an absolute value 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 we don’t have enough information, condition 1) is not sufficient.

Condition 2)
If m = 1, n = 13, then | m – n | = | 1 – 13 | = 12.
If m = 2, n = 13/2, then | m – n | = | 2 – 13/2 | = 9/2.
Since we don’t have a unique solution, condition 2) is not sufficient.

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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[GMAT math practice question]

In the xy-plane, is the triangle that connects the 3 different points A(3,4), B(p,q), and C(r,s) a right triangle?

1) (p-3)(r-3)+(q-4)(s-4)=0
2) p=3 and 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.

Condition 2):
If p = 3, then q ≠ 4 since (p,q) is different from (3,4).
If s = 4, then r ≠ 3 since (r,s) is different from (3,4).
So, (p,q) is on the line x = 3 and (r,s) is on the line y = 4.
As these lines are perpendicular AB and AC are perpendicular, and triangle ABC is a right triangle.
Thus, condition 2) is sufficient.

Condition 1) is complicated. If you can’t figure out how to apply condition 1), CMT4(B) tells you to choose D as the answer.

Condition 1)
(p-3)(r-3)+(q-4)(s-4)=0
=> (p-3)(r-3) = -(q-4)(s-4)
=> (p-3)(r-3) / (q-4)(s-4) = -1 or (q-4)(s-4) = 0
=> {(p-3)/(q-4)} * {{r-3)/(s-4)} = -1 or q = 4 or s = 4

Case 1: (p-3)/(q-4)} * {{r-3)/(s-4)} = -1
(p-3)/(q-4) and (r-3)/(s-4) are the slopes of two sides of the triangle.
Since the product of these slopes is -1, the two sides are perpendicular and the triangle is a right triangle.

Case 2: q = 4
If q = 4, then p is not 3. Also, we must have (p-3)(r- 3) = 0. So, r = 3 since (p,q) is different from (3,4).
Thus, (p,q) lies on the line y = 4 and (r,s) lies on the line x = 3.
As these lines are perpendicular, AB and AC are perpendicular, and triangle ABC is a right triangle.

Case 3: s = 4.
A similar argument to the one used for case 2 shows that triangle ABC is a right triangle.

Thus, condition 1) is also 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.
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[GMAT math practice question]

If n is the product of 5 different prime numbers, how many factors does n have?

A. 2
B. 4
C. 8
D. 16
E. 32

=>

Since p, q, r, s, t are different prime factors of n, we have n = p*q*r*s*t = p^1q^1r^1s^1t^1.
The number of factors of n is (1+1)(1+1)(1+1)(1+1)(1+1) = 2*2*2*2*2 = 32.
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[GMAT math practice question] 7.17/trend 7.24

2. (number property) The product of 4 consecutive odd integers is negative. What is the largest odd integer?
1) The smallest integer is negative.
2) The third smallest integer is negative.

=>

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 of the conditions on their own first.

Let the integers be 2n – 3, 2n – 1, 2n + 1 and 2n +3.
Since (2n-3)(2n-1)(2n+1)(2n+3) < 0, either one of the integers is negative, or three of the integers is negative. There are two possible lists of integers: -1, 1, 3, 5 and -5, -3, -1, 1.

Condition 1)
The largest odd integers in the two possible lists are 5 and 1.
Since we don’t have a unique answer, condition 1) is not sufficient.

Condition 2)
If the third smallest integer is negative, then the integers are -5, -3, -1, 1.
The largest integer is 1.
Since we have a unique answer, condition 2) is sufficient.

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: Overview of GMAT Math Question Types and Patterns on the GMAT  [#permalink]

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chetan2u wrote:
ruhibhatia wrote:
MathRevolution , can you please explain the solution of the below question.

When both positive integers a and b are divided by 7, both have a remainder of 4. What is the remainder when ab3 is divided by 7?
A. 6
B. 5
C. 4
D. 3
E. 2

Can we assume that a and b and single digit integers, since abc3 is a three-digit integer?

Hi ruhi,
there are two things which ab3 can mean..a three digit number or a*b*3..
1)if ab3 is three digit number, as you have said a and b should be a single digits..
so a and b have to be 4, as the next number to leave a remainder of 4 would be 11, which is not a single digit number..
so our number is 443.. and remainder when 443 is divided by 7 is 6..
2)if ab3 actually meant a*b*3...
remainder will be 4*4*3=48..
48 when divided by 7 leaves a remainder 6..

Hope it helps u

Hi chetan2u,

Isn't the remainder when 443 is divided by 7 is 2? Your post says 6. Could you please reconfirm?
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[Math Revolution GMAT math practice question]

If x is not 0, is x>1?

1) x/|x|<x
2) x=|x|

=>

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 of the conditions on their own first.

Condition 1)
If x > 0, x/|x| < x is equivalent to x/x < x or x > 1.
If x < 0, x/|x| < x is equivalent to x/(-x) < x or x > -1, and we have -1 < x < 0.
In inequality questions, the law “Question is King” tells us that if the solution set of the question does not include the solution set of a condition, then the condition is not sufficient.
Since the solution of the question does not include that of condition 1), condition 1) is not sufficient.

Condition 2)
x=|x| is equivalent to x ≥ 0.
Since x is not 0, we must have x > 0.
Since the solution set of the question does not include the solution set of condition 2), condition 2) is not sufficient.

Conditions 1) & 2)
Condition 1) yields x > 1 or -1 < x < 0, while condition 2) yields x > 0.
Thus, applying both conditions together yields x > 0.
Since the solution set of the question does not include the solution set of both conditions, taken together, they are not sufficient.

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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[Math Revolution GMAT math practice question]

After t seconds, the height of a ball from the ground is given by the equation h =-9.8t^2+ft+g (f and g are constants). If the ball is at its maximum height, then t=?

1) f = 10
2) g = 10

=>

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.

Y = aX^2 + bX + c has a maximum or a minimum when X = -b/2a.
Thus h =-9.8t^2+ft+g has a maximum when t = f/(2*9.8).
Condition 1) is sufficient on its own.

Condition 2) is not sufficient since it gives us no information about the value of f.

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[Math Revolution GMAT math practice question]

What is the remainder when 7^8 is divided by 100?

A. 1
B. 2
C. 3
D. 4
E. 5

=>

The remainder when 7^8 is divided by 100 is equal to the final two digits of 7^8.
Now, 7^1 = 7, 7^2 = 49, 7^3 = 343, and 7^4 = 2401.
So, the final two digits of 7^n have period 4:
The tens digits are 0 -> 4 -> 4 -> 0
and the units digits are 7 -> 9 -> 3 -> 1.
It follows that the tens and units digits of 7^8 are 0 and 1, respectively.
Therefore, the remainder when 7^8 is divided by 100 is 1.

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1
[Math Revolution GMAT math practice question]

Is 0 between x and y?

1) x-y>0
2) x^2-y^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.

If we modify the question, it asks if xy < 0. Since the two conditions do not give us enough information to determine the sign of xy, both conditions together are not sufficient, and the answer is E.

Since we have 2 variables (x and y) and 1 equation, D is most likely to be the answer. So, we should consider each of the conditions on its own first.

Conditions 1) & 2)
If x = 2 and y = -1, then 0 is between x and y.
If x = 2 and y = 1, then 0 is not between x and y.
Since we don’t have a unique solution, both conditions together are not sufficient.

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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[Math Revolution GMAT math practice question]

When 2 numbers are selected from the integers from 1 to 21 inclusive, what is the probability that the 2 selected numbers are prime numbers?

A. 2/15
B. 1/5
C. 1/3
D. 4/15
E. 2/5

=>

There are 8 prime numbers between 1 and 21, inclusive: 2, 3, 5, 7, 11, 13, 17 and 19.
So, there are 8C2 ways of selecting 2 numbers from these 8 prime numbers.
There are 21C2 ways of selecting 2 numbers from the 21 numbers from 1 to 21, inclusive.
Thus, the probability that the 2 selected numbers are prime numbers is 8C2 / 21C2 = ( 8*7 / 1*2) / ( 21*20 / 1*2 ) = 8*7 / 21*20 = 2/15.

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[Math Revolution GMAT math practice question]

If (n+2)!/n!=90, then n=?

A. 8
B. 9
C. 10
D. 11
E. 12

=>

(n+2)!/n!= (n+2)(n+1) = 90 = 10*9.
Thus, n + 2 = 10 or n = 8.

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[Math Revolution GMAT math practice question]

If n is a 2-digit positive integer and its tens digit is 4 times its units digit, what is the value of n?

1) The tens digit of n is 8.
2) The sum of the tens digit and the units digit of n is 2-digit 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.

n = 10a + b and a = 4b
The possible pairs ( a, b ) are ( 4, 1 ) and ( 8, 2 ) by the original condition.
Thus, n = 41 or n = 82.

Since we have 3 variables and 2 equations, D is most likely to be the answer. So, we should consider each of the conditions on its own first.

Condition 1)
Since a = 8, we must have n = 82.
Thus, condition 1) is sufficient.

Condition 2)
Since 4 + 1 = 5 is not a 2-digit integer and 8 + 2 = 10 is a 2-digit integer, n = 82.
Thus, condition 2) is sufficient.

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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[Math Revolution GMAT math practice question]

Is ab>bc?

1) abc=0
2) a>c

=>

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.

When we modify the question, ab > bc is equivalent to ab – bc > 0 or b(a-c) > 0. Even though we know a – c > 0 from condition 2), we don’t know if b is positive or negative. Thus, both conditions together are not sufficient.

Conditions 1) & 2):
If a = 1, b =1 and c = 0, then ab > bc and the answer is ‘yes’.
If a = 1, b =-1 and c = 0, then ab < bc and the answer is ‘no’.
Since we don’t have a unique solution, both conditions together are not sufficient.

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[Math Revolution GMAT math practice question]

If x, y, and z are positive integers, x=?

1) y=x+1 and z=x+3
2) x, y, and z are prime 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.

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)
Since y = x + 1, z = x + 3 and x, y, z are prime numbers, the only possibility is that x = 2, y = 3, and z = 5 because if x is an odd number, then y and z are two different even numbers, which cannot both be prime numbers.
Thus, 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)
There are many different possible values for x, y and z, including x = 1, y = 2, z = 4 and x = 2, y = 3, z = 5. Therefore, we don’t have a unique solution, and condition 1) is not sufficient.

Condition 2)
There are many different possibly values for x, y and z, including x = 2, y = 3 and z = 5, and x = 3, y = 5 and z = 7. So, we don’t have a unique solution, and condition 2) is not 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.
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[Math Revolution GMAT math practice question]

If (2/3)^n=(9/4)^2, what is the value of n?

A. -4
B. -2
C. 0
D. 2
E. 4

=>

(9/4)^2 = ((3/2)^2)^2 = (3/2)^4 = (2/3)^-4.
Thus, n = -4.

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[Math Revolution GMAT math practice question]

If x and y are prime numbers, how many factors has x^2y^2?

1) xy=10
2) x+y is 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.

If x and y are different prime numbers, then xy has (2+1)(2+1) = 9 factors.
If x and y are the same prime number, then xy has 4+1 = 5 factors.

Condition 1)
Since x and y are prime numbers and xy = 10, either x = 2 and y = 5, or x = 5 and y = 2.
So, x and y are different prime numbers. Thus, condition 1) is sufficient.

Condition 2)
Since x and y are prime numbers and x + y is odd, one of them is even and the other one is odd.
So, x and y are different prime numbers. Thus, condition 2) is sufficient.

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1
[Math Revolution GMAT math practice question]

Is x|y|=|xy|?

1) x>0
2) y>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.

Modifying the question:
x|y|=|xy|
=> x|y|-|xy| = 0
=> x|y|-|x||y| = 0
=> |y|(x-|x|) = 0
=> |y|=0 or x-|x| = 0
=> y=0 or x=|x|
=> y=0 or x≥0

Condition 1) is sufficient.
Condition 2) is not sufficient since it tells us nothing about the value of x.

_________________
Math Revolution GMAT Instructor V
Joined: 16 Aug 2015
Posts: 8445
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]

Alice, Bob, Cindy, Dave and Eddie joined a three-person-a-side basketball tournament. In how many ways can be the three starters be chosen?

A. 5
B. 6
C. 8
D. 9
E. 10

=>

The number of ways of choosing the three starters is the number of ways of choosing three people from a set of five people, where order does not matter and repetition is not allowed.
The number of possible selections of three starters is:
5C3 = 5C2 = (5*4)/(1*2) = 10.

_________________ Re: Overview of GMAT Math Question Types and Patterns on the GMAT   [#permalink] 30 Aug 2018, 19:02

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