[Math Revolution GMAT math practice question]

If x and y are integers, is x2-y2 an even integer?

1) x^3-y^3 is an even integer
2) x+y is an even integer

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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 whether x^2-y^2 is even is equivalent to asking whether (x+y)(x-y) is an even integer.

Condition 2) is sufficient since x^2-y^2 = (x+y)(x-y) is an even integer if x + y is an even integer.

Condition 1)
In order for x^3-y^3 to be an even integer, x and y must both have the same parity.
There are only two cases to consider: both x and y are even integers or both are odd integers.
Since x and y have the same parity, x – y is always an even integer.
Thus, x^2-y^2 is an even integer. Condition 1) is sufficient.

Therefore, D is the answer.
Answer: D
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[Math Revolution GMAT math practice question]

Is ab<0?

1) |a+b| = - ( a + b )
2) |a+b| + 1 = |a| + |b|

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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.

You should remember that the inequality |x+y| < |x| + |y| is equivalent to the inequality xy < 0.

Condition 2) tells us that |a+b| + 1 = |a| + |b|. Thus, |a + b| < |a| + |b| and ab < 0.
Thus, condition 2) is sufficient.

Condition 1)
If a = -2 and b = 1, then the answer is ‘yes’.
If a = -1 and b = -1, then the answer is ‘no’.
Since it does not yield a unique solution, condition 1) is not sufficient.

Therefore, the answer is B.
Answer: B
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[Math Revolution GMAT math practice question]

n is a 3 digit integer of the form ab6. Is n divisible by 4?

1) a+b is an even integer
2) ab is an odd integer.

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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.

We can determine whether a number is divisible by 4 from its final two digits.
Numbers with the final digits 16, 36, 56, 76 and 96 are divisible by 4 and those with final digits 06, 26, 46, 66 and 88 are not divisible by 4. Thus, asking whether n is divisible by 4 is equivalent to asking whether b is odd.

Since it implies that both a and b are odd integers, condition 2) is sufficient.

Condition 1)
There are two cases to consider.
If a is an even integer and b is an odd integer, the answer is ‘yes’.
If a is an odd integer and b is an even integer, the answer is ‘no’.
Since it does not yield a unique solution, condition 1) is not sufficient.

Therefore, B is the answer.
Answer: B
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[Math Revolution GMAT math practice question]

If x and y are non-zero numbers and x≠±y, then ( x^2 + y^2 ) / ( x^2 - y^2 )=?

1) |x/y|=1/3
2) y=-3x

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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 ( x^2 + y^2 ) / ( x^2 - y^2 )= ( (x/y)^2 + 1 ) / (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 |x/y|=1/3, x/y = ±(1/3), and ( x^2 + y^2 ) / ( x^2 - y^2 )= ( (x/y)^2 + 1 ) / ( (x/y)^2 – 1 ) = ( (1/3)^2 + 1 ) / ( (1/3)^2 – 1) = (1/9 + 1)/(1/9-1) = (10/9)/(-8/9) = -10/8 = -5/4.
Condition 1) is sufficient since it gives a unique solution.


Condition 2)
Since y = -3x, x/y = -1/3, and ( x^2 + y^2 ) / ( x^2 - y^2 )= ( (x/y)^2 + 1 ) / ( (x/y)^2 – 1 ) = ( (-1/3)^2 + 1 ) / ( (-1/3)^2 – 1) = (1/9 + 1)/(1/9-1) = (10/9)/(-8/9) = -10/8 = -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.

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[Math Revolution GMAT math practice question]

A = (2-3+4)^{11}, and B = (-2+3-4)^{11}. What is the value of 2^{A+B}?

A. 1/2048
B. 1/1024
C. 1
D. 1024
E. 2048

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A + B
= (2-3+4)^{11}+ (-2+3-4)^{11}
= 3^{11}+ (-3)^{11}
= 3^{11}- 3^{11}
= 0.
Therefore,
2^{A+B} = 2^0 = 1.

Therefore, the answer is C.
Answer: C

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0 is also a multiple of 4 and 6. If we take n as 0, then 4^n+n^2+1 = 4^0+0^2+1 = 2, which is divisible by 4 and 6, but if we take n as 4 or 6, then the expression is not divisible by both 4 and 6. Hence Shouldn't E be the answer?
chetan2u sir, Am I wrong?
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[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

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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
( 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
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[GMAT math practice question]

If p, x, and y are integers, x^p/x^q=?

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

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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 x^p/x^q = x^{p-q} = x^4 = 2^4 = 16.
If x = 16, p = 5 and q = 1, then 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.

Therefore, the answer is E.
Answer: E

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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[GMAT math practice question]

If x, y, z are positive integers, is xyz64?

1) xy≥yz≥zx≥16
2) x+y+z=64

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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, z) and 0 equations, E is most likely to be the answer. So, we should consider each condition on its own first. As condition 1) includes 3 equations, we should consider it first.

Condition 1)
Since xy ≥ 16, yz ≥ 16, and zx ≥ 16, we have (xyz)^2 ≥ 16^3 or xyz ≥ 64.
Condition 1) is sufficient.

Condition 2)
If x = 20, y = 20 and z = 24, then xyz = 9600 ≥ 64 and the answer is ‘yes’.
If x = 1, y = 1 and z = 62, then xyz = 62 < 64 and the answer is ‘no’.
Since condition 2) does not yield a unique answer, condition 2) is not sufficient.

Therefore, A is the answer.
Answer: A

This is a CMT(Common Mistake Type) 4(A) question. If a question is from one of the key question areas and C should be the answer, CMT (Common Mistake Type) 4(A) of the VA (Variable Approach) method tells us that we should also check answers A and B.
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[GMAT math practice question]

Does f(x)=ax^4+bx^3+cx^2+dx+e have (x-2) as a factor?

1) f(2)=0
2) f(-2)=0

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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 statement that f(x)=ax^4+bx^3+cx^2+dx+e has (x-2) as a factor is equivalent to saying f(x) = (x-2)g(x) for some function g(x). This is equivalent to the requirement that f(2) = 0.

Thus condition 1) is sufficient.

Condition 2)
If f(x) = (x-2)(x+2)(x-1)(x+1), then f(x) has x-2 as a factor, The answer is ‘yes’.
If f(x) = x(x+2)(x+1)(x-1), then f(x) doesn’t have x-2 as a factor. The answer is ‘no’.
Since condition 2) does not yield a unique answer, condition 2) is not sufficient.

Therefore, the answer is A.
Answer: A
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