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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.

Since the last three digits 000 is a multiple of 8 and abc000 is a multiple of 8, the question “is abc000 divisible by 24?” is equivalent to “is abc000 divisible by 3?” or “is a + b + c divisible by 3?”.
Thus, condition 2) is sufficient.

Condition 1) is not sufficient as 8 is not divisible by 3.

Therefore, the correct answer is B.
Answer: 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.

Since we have 1 variable (a) 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)
\(|a| = 1 => a = ±1\).
If \(a = 1\), then \(f(a) = f(1) = 1.\)
If \(a = -1\), then \(f(a) = f(-1) = (-1)^2 = 1\).
Since it gives a unique solution, condition 1) is sufficient.

Condition 2)
\(a^2 - a = 0 => a(a-1) = 0 => a=0\) or \(a =1\)
If \(a = 0\), then\(f(a) = f(0) = 0\).
If \(a = 1\), then \(f(a) = f(1) = 1.\)
Since it does not give a unique solution, condition 2) is not sufficient.

Therefore, A is the answer.
Answer: A

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.

Condition 1) is obviously sufficient.

For condition 2), we have two cases.
If we have 14, 13, 9, 4, then only two numbers are greater than 10.
If we have 14, 11, 11, 4, then three numbers are greater than 10.
Thus condition 2) is not sufficient.

Therefore, A is the answer.

[Math Revolution GMAT math practice question]

(number property) If \(n\) is an integer between \(30\) and \(50\) inclusive, what is the value of \(n\)?

1) When \(n\) is divided by \(8\), the remainder is \(7\)
2) When \(n\) is divided by \(16\), the remainder is \(7\)

[Math Revolution GMAT math practice question]

(number property) If \(p, q\) and \(r\) are prime, with \(p<q<r, p=?\)

\(1) (pq)^3=216\)
\(2) (pr)^3=1000\)

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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 (n) and \(0\) equations, D is most likely to be the answer. So, we should consider each condition on its own first.

Condition 1)
We can express \(n = 8k+7\) for some integer \(k\).
If \(k = 3\), then \(n = 31.\)
If \(k = 4\), then \(n = 39.\)
Since we don’t have a unique solution, condition 1) is not sufficient.

Condition 2)
We can express \(n = 16m+7\) for some integer \(m\).
If \(m = 2\), then \(n = 39.\)
If \(m = 1\), then \(n = 23\) and \(n < 30.\)
If \(m = 3,\) then \(n = 55\) and \(n > 50.\)
Thus \(n = 39\) is the unique solution and condition 2) is sufficient.

Therefore, B is the answer.
Answer: B

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 math practice question]

(number property) If \(n\) is an integer greater than \(1\), what is the value of \(n\)?

1) \(n\) is a prime number
2) \(\frac{(n+2)}{n}\) is an 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.

Since we have \(3\) variables (\(p, q\) and \(r\)) 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)

\((pq)^3=216\)
\(=> p^3q^3=2^33^3\)
\(=> p = 2\) and \(q = 3\), since \(p\) and \(q\) are prime numbers with \(p < q.\)

\((pr)^3=1000\)
\(=> p^3r^3=2^35^3\)
\(=> p = 2\) and \(r = 5\), since \(p\) and \(r\) are prime numbers with \(p < r.\)

While we have checked both conditions together, we have shown that conditions 1) and 2) are equivalent to each other in terms of \(p\). So, each condition is sufficient by Tip 1).
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.

Therefore, the answer is D.
Answer: D

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 math practice question]

(inequality) If \(x\) and \(y\) are positive, is \(1<x<y\)?

\(1) √x<x<y\)
\(2) 1<√x<y\)

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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 (n) 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)
Since there are many prime numbers, condition 1) is not sufficient.

Condition 2)
If \(n = 1\), then \(\frac{(n+2)}{n} = 3\) is an integer.
If \(n = 2,\) then \(\frac{(n+2)}{n} = 2\) is an integer.
Since we don’t have a unique solution, condition 2) is not sufficient.

Conditions 1) & 2)
If \(n = 2\), then \(\frac{(n+2)}{n} = 2\) is an integer.
If \(n = 3\), then \(\frac{(n+2)}{n} = \frac{5}{2}\) is not integer.
If \(n\) is a prime number bigger than \(2\), \(\frac{(n+2)}{n}\) is not an integer.
Thus \(n = 2\) is the unique solution and both conditions together are sufficient.

Therefore, C is the answer.
Answer: C

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 math practice question]

(inequality) Is \(x^3-4x>0？\)

\(1) x>2\)
\(2) x>-2\)

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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 \(2\) 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)
Since \(\sqrt{x}<x<y\) and \(1<\sqrt{x}<y\), we have \(1<\sqrt{x}<x<y.\) Both conditions together are sufficient.

Since this question is an inequality 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 \(\sqrt{x}<x,\) we have \(x > 1.\)
Thus, \(1<\sqrt{x}<x<y\) and condition 1) is sufficient.

Condition 2)
If \(x = 2\) and \(y = 3\), then the answer is ‘yes’.
If \(x = 4\) and \(y = 3\), then the answer is ‘no’
Thus, condition 2) is not sufficient, since it does not yield a unique solution.

Therefore, the correct answer is A.
Answer: A

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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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.

\(x^3-4x>0\)
\(=> x(x^2-4)>0\)
\(=> x(x+2)(x-2)>0\)
\(=> -2<x<0\) or \(x > 2\)

Since we have \(1\) variable (\(x\)) and \(0\) equations, D is most likely to be the answer. So, we should consider the conditions on their own first.

Condition 1)
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
Since the solution set of the question, \(-2<x<0\) or \(x > 2\), includes the solution set of condition 1), \(x > 2\), condition 1) is sufficient.

Condition 2)
The solution set of the question, \(-2<x<0\) or \(x > 2\), does not include the solution set of condition 2), \(x > -2\), so condition 2) is not sufficient.

Therefore, A is the answer.
Answer: A

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 math practice question]

(number properties) If \(x\) and \(y\) are positive integers, is \(\sqrt{15xy}\) an integer?

1) \(xy\) is a multiple of \(15\)
2) \(x\) and \(y\) are prime numbers

[Math Revolution GMAT math practice question]

(functions) In the x-y plane, line l passes through points \((-1,-1)\) and \((3,k)\). What is the value of \(k\)?

1) The y-intercept of line l is \(1\)
2) The slope of line l is \(2\)

From statement 1:

y-intercept, which is C in y = mx+c. Given C = 1.
slope m = \(\frac{y_2-y_1}{x_2-x_1}\) = \(\frac{k+1}{4}\)
y = \(\frac{k+1}{4}\)*x + 1.
using any if the points (-1,-1) or (3,k) gives k = 7.
Sufficient.

From statement 2:

Slope, m = 2.
m = \(\frac{k+1}{4}\) = 2.
Solving gives k as 7.
Sufficient.

D is the answer.

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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 \(2\) 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)
When we consider both conditions together, there are two sets of possible values of \(x\) and \(y: x = 3, y = 5\) and \(x = 5, y = 3\). In both cases, \(xy = 15\), so
\(\sqrt{15xy} = \sqrt{15*3*5} = \sqrt{225} = 15\) is an integer.
Thus, both conditions together are sufficient.

Since this question is a statistics 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)
If \(x = 3\) and \(y = 5\), then \(\sqrt{15xy} = \sqrt{15*3*5} = \sqrt{225} = 15\) is an integer.
If \(x = 6\) and \(y = 5\), then \(\sqrt{15xy} = \sqrt{15*6*5} = \sqrt{450} = 15\sqrt{2}\) is not an integer.
Since we don’t have a unique answer, condition 1) is not sufficient by CMT (Common Mistake Type) 2.

Condition 2)
If \(x = 3\) and \(y = 5\), then \(\sqrt{15xy} = \sqrt{15*3*5} = \sqrt{225} = 15\) is an integer.
If \(x = 2\) and \(y = 5\), then \(\sqrt{15xy} = \sqrt{15*2*5} = \sqrt{150} = 5\sqrt{6}\) is not an integer.
Since we don’t have a unique answer, condition 2) is not sufficient by CMT (Common Mistake Type) 2.

Therefore, C is the answer.
Answer: C

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.