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[GMAT math practice question]

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

1) \(m\) and \(n\) are consecutive
2) \(m^2 – n^2 = m + n\)

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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 “\(m^2-n^2\) is divisible by \(4\)” means that \((m+n)(m-n)\) is divisible by \(4.\) This is equivalent to the requirement that \(m\) and \(n\) are either both even integers or both odd integers.

Since condition 2) tells us that both \(m\) and \(n\) are odd integers, condition 2) is sufficient.

Condition 1)
The square of an odd integer \((2a+1)^2 = 4a^2 + 4a + 1 = 4(a^2 + a) + 1\) has remainder \(1\) when it is divided by \(4\).
The square of an even integer \((2b)^2 = 4b^2\) has remainder \(0\) when it is divided by \(4\).
Thus, if “\(m^2+n^2\) has remainder \(2\) when it is divided by \(4\)”, both \(m\) and \(n\) must be odd integers.
Condition 1) is sufficient.

Therefore, D is the answer.
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.

[GMAT math practice question]

(number property) If \(p\) and \(q\) are prime numbers, what is the number of factors of \(6pq\)?

1) \(p\) and \(q\) are different

2) \(p<3<q\)

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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.

Condition 1)
The gcd of two consecutive integers is always \(1\).
Thus, condition 1) is sufficient.

Condition 2)
If \(m^2–n^2 = m+n\), then \((m+n)(m-n)=m+n\) and \(m-n = 1\) since \(m+n ≠0\).
This implies that \(m\) and \(n\) are consecutive integers, and their \(gcd\) is \(1\).
Condition 2) is sufficient since it yields a unique answer.

Therefore, D is the answer.
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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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.

Recall that if \(n = p^aq^br^c\), where \(p, q\) and \(r\) are different prime numbers, and \(a, b\) and \(c\) are non-negative integers, then \(n\) has \((a+1)(b+1)(c+1)\) factors.

Condition 2)
We must have \(p = 2\) since \(p\) is prime and \(p < 3.\)

The prime factorization of \(6pq\) is \(2*3*p*q = 2^2*3*q\) since \(q\) is prime and \(q > 3\).

The number of factors of \(2^2*3*q is (2+1)(1+1)(1+1) = 12\). Condition 2) is sufficient since it yields a unique answer.

Condition 1)
If \(p = 2\) and \(q = 3\), then \(6pq = 2^2*3^2\) and the number of factors is \((2+1)(2+1)= 9.\)

If \(p = 5\) and \(q = 7\), then \(6pq = 2*3*5*7\) and the number of factors is \((1+1)(1+1)(1+1)(1+1) = 8.\)

Condition 1) is not sufficient since it does not yield a unique solution.

Therefore, B is the answer.
Answer: B

[GMAT math practice question]

(number properties) \(m\) and \(n\) are positive integers greater than \(6\). What is the value of \(m + n\)?

\(1) m*n = 504\)

\(2) m\) and \(n\) are multiples of \(6\)

[GMAT math practice question]

(number properties) \(m * n = 2145\), where \(m\) and \(n\) are positive integers. What is the value of \(m + n\)?

1) \(m\) and \(n\) are two-digit integers.

2) \(m\) and \(n\) both have remainder \(3\) when they are divided by \(4\)

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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 (\(m\) 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)
Condition 1) allows us to write \(m = 6a\) and \(n = 6b\), where \(a\) and \(b\) are integers greater than \(1\).
So, \(m*n = 6a*6b = 36*a*b = 504\).
This yields \(ab = 14\). So, \(a = 2\) and \(b = 7\) or \(a = 7\) and \(b = 2\).
Thus, \(m=12\) and \(n=42\) or \(m=42\) and \(n=12\), and we obtain the unique answer \(m+n = 54\).
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)
\(m*n = 504 = 2^3*3^2*7\)
If \(m = 12\) and \(n = 42\), then \(m + n = 54.\)
If \(m = 8\) and \(n = 63\), then \(m + n = 71\).
Condition 1) is not sufficient since it does not yield a unique answer.

Condition 2)
If \(m = 12\) and \(n = 12\), then \(m + n = 24\).
If \(m = 12\) and \(n = 24\), then \(m + n = 36\).
Condition 2) is not sufficient since it does not yield a unique answer.

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]

(geometry) What is the area of triangle ABC?

1) Triangle ABC has two sides of lengths 3 and 4
2) Triangle ABC is a right triangle

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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 (\(m\) and \(n\)) and \(1\) equation (\(mn=2145\)), D is most likely to be the answer.

Condition 1)
\(m * n = 2145 = 3*5*11*13.\)

Since \(m\) and \(n\) are two-digit numbers, there are four cases to consider:

i) \(m=33 (= 3*11)\) and \(n=65 (= 5*13)\)

ii) \(m=39 (=3*13)\) and \(n=55 (=5*11)\)

iii) \(m=65 (=5*13)\) and \(n=33 (= 3*11)\)

iv) \(m=55 (=5*11)\) and \(n=39 (=3*13)\)

So, there are two possible values of \(m + n\), which are \(98\) and \(94.\)

Condition 1) is not sufficient since it does not yield a unique answer.

Condition 2)
If m = 39 and n = 55, then m + n = 94.
If m = 3 and n = 715(=5*11*13), then m + n = 718.
Condition 2) is not sufficient since it does not yield a unique answer.

Conditions 1) & 2)
\(m * n = 2145 = 3*5*11*13.\)

Condition 1) gives rise to the following four cases for the values of \(m\) and \(n\):

i) \(m=33 (= 3*11)\) and \(n=65 (= 5*13)\)

ii) \(m=39 (=3*13)\) and \(n=55 (=5*11)\)

iii) \(m=65 (=5*13)\) and \(n=33 (= 3*11)\)

iv) \(m=55 (=5*11)\) and \(n=39 (=3*13)\)

Of these, only
\(m=39\) and \(n=55\), and \(m=55\) and \(n=39\) satisfy condition 2)

So, we have a unique answer \(m+n=94\).

Thus, both conditions together are 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]

(function) The parabola \(y=f(x)=a(x-h)^2+k\) lies in the \(x-y\) plane. What is the value of \(k\)?

1) \(y=f(x)\) passes through \((1,0)\) and \((3,0)\).

2) \(y=f(x)\) passes through \((2,1)\) and no \(y\)-value is greater than \(1\).

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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 a triangle has 3 variables in geometry, 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)



There are two right triangles with sides of lengths 3 and 4 as shown above.
Thus, there are two possible areas: \((\frac{1}{2})*4*3 = 6\) and \((\frac{3}{2}) √7.\)

Since the conditions don’t yield a unique answer when applied together, they are not sufficient.

Therefore, E is the answer.
Answer: E

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.

[GMAT math practice question]

(number property) If \(a, b\), and \(c\) are integers, is \(a+b+c\) an even integer?

1) \(a^2+b^2\) is an even integer
2) \(b^2+c^2\) 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.



The question asks for the minimum or maximum value for the function.
Condition 2 is sufficient since it provides the maximum value for the function.

Condition 1)



The parabolas drawn above both pass through the points (1,0) and (3,0). It is obvious that we don’t have a unique maximum or minimum function value. Condition 1) is not sufficient.

Therefore, B is the answer.
Answer: 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.

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 \(a = 1, b = 1\) and \(c = 1\), then \(a + b + c = 3\) is not an even integer and the answer is ‘no’.

If \(a = 2, b = 2\) and \(c = 2\), then \(a + b + c = 6\) is an even integer and the answer is ‘yes’.

Since the conditions don’t yield a unique answer when applied together, they are not sufficient.

Therefore, E is the answer.
Answer: E

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.

[GMAT math practice question]

(absolute value) \(y≠0\). What is the value of \(\frac{x}{y}\)?

\(1) x^2-6xy+9y^2=0\)

\(2) |x-3|+|y-1|=0\)

[GMAT math practice question]

(number properties) If \(n\) is a positive integer, what is the value of \(n\)?

1) \(n(n-1)\) is a prime number

2) \(n(n+1)\) has \(4\) factors