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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 should simplify conditions if necessary.

Condition 1)
If \(a, b\) and \(c\) are the lengths of the three sides of a triangle, we must have

\(a > b – c\) since the length of each side of a triangle is always greater than the difference between the lengths of the other two sides. Thus, condition 1) is sufficient.

Condition 2)
We can assume \(a, b\) and \(c\) are the sides of a right triangle. The above reasoning tells us that \(a > b – c\). Thus, condition 2) is also sufficient.

Therefore, D is the answer.
Answer: D

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).

[GMAT math practice question]

(number properties) What is the greatest common divisor of positive integers \(m\) and \(n\)?

1) \(m\) and \(n\) are different prime numbers

2) \(m\) and \(n\) are consecutive integers

Hi, GloryBoy92

GMAT math is a logical test rather than a test of general math skills. You won't be able to hit Q49-51 by solving the questions with conventional methods, nor will you master the logic of GMAT Math with conventional methods. We teach our own unique approaches (Variable approach for DS and IVY approach for PS) that will shorten your study hours and help you hit your target score. It's not impossible to hit even the high score of Q50, as long as you carefully follow our study instructions.

It is suggested that you visit math revolution site and see details about that.

You can take a look at the attachment.
The case with 2 variables and 2 equations is very rare. D is most likely to be the answer in that case. However, I am not sure you counted the numbers of equations and variables correctly.

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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 should simplify the conditions if necessary.

Modifying the question:
For \(abcd + abc + ab + a = a(bcd+bc+b+1)\) to be even, either a must be even or \(bcd + bc + b + 1\) must be even.

Condition 2):
If \(bcd\) is an odd integer, then \(b, c\) and \(d\) are odd integers. This implies that \(bc\) is odd, and \(bcd + bc + b + 1\) is an even integer. Condition 2) is sufficient.

Condition 1)
If \(a = b = c = d = 1\), then \(abcd + abc + ab + a = 1 + 1 + 1 + 1 = 4\), which is an even integer, and the answer is ‘yes’.
If \(a = b = c = 1\) and \(d = 2\), then \(abcd + abc + ab + a = 2 + 1 + 1 + 1 = 5\), which is an odd integer, and the answer is ‘no’.
Condition 1) is not sufficient since it doesn’t yield a unique solution.

Therefore, B is the answer.
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.

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

Condition 1)
\(m\) and \(n\) have a unique common divisor since \(1\) and \(m\) are the only factors of \(m\) and, \(1\) and \(n\) are the only factors of \(n\). This tells us that \(gcd(m,n)=1\) and condition 1) is sufficient.

Condition 2)
Since the greatest common divisor of consecutive integers is \(1\), condition 2) is sufficient.

Therefore, D is the answer.
Answer: D

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).

[GMAT math practice question]

(number properties) If \(a\) and \(b\) are positive integers, is \(a^2-b^2\) divisible by \(4\)?

1) \(a+b\) is divisible by \(4\)

2) \(a^2+b^2\) has remainder \(2\) when it is divided by \(4\)

[GMAT math practice question]

(number properties) \(m\) is an odd integer. If \(m^3n^4=432\), what is the value of \(n\)?

1) \(n\) is positive.

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.

The first step of the VA (Variable Approach) method is to modify the original condition and the question. We then recheck the question. We should simplify the conditions, if necessary.

Condition 1)
If \(a + b\) is divisible by \(4\), then \(a^2 – b^2 = (a+b)(a-b)\) is divisible by \(4\).

Thus, condition 1) is sufficient.

Condition 2)
The squares of \(1, 2, 3, 4, …\) are \(1, 4, 9, 16, …\), respectively and they have remainders of \(1, 0, 1, 2, …\), respectively, when they are divided by \(4\).

Thus, if \(a^2 + b^2\) has remainder \(2\) when it is divided by \(4\), both \(a\) and \(b\) are odd integers.

This implies that both \(a + b\) and \(a – b\) are even integers, and \(a^2 – b^2 = ( a + b )( a – b )\) is divisible by \(4\).
Thus, condition 2) is sufficient too.

Therefore, D is the answer.
Answer: D

This question is a CMT4(B) question: condition 1) is easy to work with and condition 2) is difficult to work with. For CMT4(B) questions, D is most likely to be the answer.

[GMAT math practice question]

(statistics) The data set \(X\) has \(6\) elements. Its mean is \(0\) and its standard deviation is \(d\), where \(d\) is not zero. When we add a new data element \(x\) to the set \(X\), \(D\) is the standard deviation of the new set of \(7\) elements. Is \(D < d\)?

\(1) |x| < d\)

\(2) x = 0\)

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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.

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 \(m\) is an odd integer, \(n\) is a positive integer and \(432 = 3^3*2^4\), the unique solution pair is \(m = 3\) and \(n = 2\).
Thus, both conditions are sufficient, when applied together.

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)
\(m = 3\) and \(n = 2\) is a pair of solutions.
\(m = 1\) and \(n = 4√432\) is another pair of solutions.
Since condition 1) doesn’t yield a unique solution, it is not sufficient.

Condition 2)
\(m = 3\) and \(n = 2\) is a pair of solutions.
\(m = 3\) and \(n = -2\) is another pair of solutions.
Since condition 2) doesn’t yield a unique solution, it is not sufficient.

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.

[GMAT math practice question]

(number properties) For positive integers \(m\) and \(n\), is \(3+3^2+3^3+….+3^{mn+1}\) divisible by \(6\)?

1) \(m^2 + n^2\) has remainder \(1\) when it is divided by \(4\).

2) \(\frac{m}{n}\) 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.

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

Set X={a1, a2, ……, a6}, mean=0 and standard deviation=d. The new set = {a1, a2, ……, a6, x} has standard deviation D. Recall that the standard deviation reflects the distance between each element of the data set and the data set’s average,

Conditions 1)
Since the distance between x and the mean of the set X is less than d, the standard deviation D of the new set is less than d.
Thus, condition 1) is sufficient.

Condition 2)
Since the distance between 0 and the mean of the set X is less than d, the standard deviation D of the new set is less than d.
Thus condition 2) is sufficient.

Therefore, D is the answer.
Answer: D

This question is a CMT4(B) question: condition 2) is easy to work with and condition 1) is difficult to work with. For CMT4(B) questions, D is most likely to be the answer.

[GMAT math practice question]

(statistics) If \(x > y,\) then what is the median of \(x, y, 9\) and \(9\)?

1) The average (arithmetic mean) of \(x\) and \(y\) is \(9\).

2) The average of \(x, y\) and \(18\) is \(12\).

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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 should simplify the conditions if necessary.

The statement that \(3+3^2+3^3+….+3^{mn}\) is divisible by \(6\) is equivalent to the statement that \(3+3^2+3^3+….+3^{mn}\) is divisible by \(2\), because each of \(3, 3^2,3^3, …\)and \(3^{mn}\) is divisible by \(3\). It is also equivalent to the statement that \(mn\) is an odd number or (\(mn+1\)) is an even number.

Condition 1)
The squares of \(1, 2, 3, 4, …\) are \(1, 4, 9, 16, …\), respectively, and they have remainders of \(1, 0, 1, 2, …\) , respectively, when they are divided by \(4\).
Thus, if \(m^2 + n^2\) has a remainder of \(2\) when it is divided by \(4\), both \(m\) and \(n\) are odd integers.
It follows that \(mn\) is an odd number, and condition 1) is sufficient.

Condition 2)
If \(m = 1\) and \(n = 1\), then \(3^1 + 3^2 = 12\) is an even number, and the answer is ‘yes’.
If \(m = 2\) and \(n = 1\), then \(3^1 + 3^2 + 3^3 = 37\) is an odd number, and the answer is ‘no’.
Since condition 2) doesn’t yield a unique solution, it is not sufficient.

Therefore, A is the answer.
Answer: A

=>

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 should simplify the conditions, if necessary.

Condition 1)
Since \(x > y\) and the average of \(x\) and \(y\) is \(9\), we have \(x > 9 > y.\)

Thus, the median of \(x, 9, 9\) and \(y\) is \(9\).

Since condition 1) yields a unique solution, it is sufficient.

Condition 2)
Since \(\frac{( x + y + 18 )}{3} = 12\) or \(x + y + 18 = 36\), the average of \(x\) and \(y\) is \(9.\)

Condition 2) is sufficient by the same reasoning as condition 1).

Therefore, D is the answer.
Answer: D

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).

This question is a CMT4(B) question: condition 1) is easy to work with and condition 2) is difficult to work with. For CMT4(B) questions, D is most likely to be the answer.

[GMAT math practice question]

(number properties) If \(k, m\) and \(n\) are positive integers, what is the value of \(kmn\)?

\(1) k^4mn = 240\)

\(2) m = 3\)

[GMAT math practice question]

(number properties) \(p\) and \(q\) are different positive integers. What is the remainder when \(p^2 + q^2\) is divided by \(4\)?

1) \(p\) and \(q\) are prime numbers.

2) \(p\) and \(q\) are not consecutive integers.