The Ultimate Q51 Guide [Expert Level]

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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.
Visit https://www.mathrevolution.com/gmat/lesson for details.

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.

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)
We have \(m = 5a + 4\) and \(n = 5b + 2\) for some integers \(a\) and \(b\).

Then we have \(mn = (5a+4)(5b+2) = 25ab + 10a + 20b + 8 = 5(5ab+2a+4b+1)+3\) and \(mn\) has a remainder \(3\) when it is divided by \(5.\)

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)
We have \(m = 5a + 4\) from condition 1)
If we have \(m = 4\) and \(n = 1,\) then \(mn = 4\) has a remainder \(4\) when it is divided by \(5.\)

If we have \(m = 4\) and \(n = 2\), then \(mn = 8\) has a remainder \(3\) when it is divided by \(5\).

Since condition 1) does not yield a unique solution, it is not sufficient.

Condition 2)
We have \(m = 5b + 2\) from condition 1)
If we have \(m = 1\) and \(n = 2\), then \(mn = 2\) has a remainder \(2\) when it is divided by \(5.\)

If we have \(m = 4\) and \(n = 2,\) then \(mn = 8\) has a remainder \(3\) when it is divided by \(5.\)
Since condition 2) does not 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.
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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 “does the graph of \(y=ax^2+c\) intersect the x-axis” is equivalent to asking “does the equation \(ax^2+c = 0\) have a root”.
Note that the statement “\(ax^2 + bx + c = 0\) has a root” is equivalent to \(b^2-4ac ≥ 0.\)
Thus, the question asks if \(-4ac ≥ 0,\) or \(ac ≤ 0\), since \(b = 0\) in this problem.

When we consider both conditions together, we obtain \(ac > 0\) and the answer is “no”, since \(a > 0\) and \(c > 0.\)
Since ‘no’ is also a unique answer by CMT (Common Mistake Type) 1, both conditions together are sufficient.

Note: Neither condition on its own provides enough information for us to determine whether \(ac ≤ 0.\)

Therefore, C is the answer.
Answer: C
_________________

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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.
Visit https://www.mathrevolution.com/gmat/lesson for details.

Since there are \(6\) bags, we have many variables and \(0\) equations, and E is most likely 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)
Assume \(a\) and \(b\) are the numbers of beads that Adam and Betty have, respectively.
Since Adam chooses \(3\) bags and Betty chooses \(2\) bags, one bag is not chosen by Adam or Betty.
Since we have \(a = 2b, a + b = 2b + b = 3b\), then \(a + b\) is a multiple of \(3.\)

\(18 + 19 + 21 + 23 + 25 + 34 = 140\) has a remainder \(2\) when \(140\) is divided by \(3\). When we subtract the number of beads in \(1\) of the \(6\) bags from \(140\), it must be a multiple of \(3\) and only \(23\) satisfies this condition, since \(140 – 23 = 117\), which is a multiple of \(3.\)

\(23\) is the number of broken beads.
Since both conditions together yield a unique solution, they 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)
There are \(6\) possibilities for the bag with broken beads.
Since condition 1) does not yield a unique solution, it is not sufficient.

Condition 2)
Since condition 2) doesn’t give us any information about the number of broken beads, it is obviously not sufficient.

Therefore, C is the answer.
Answer: C

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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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)
\(x△1 = (x-1)^2 + (1+1)^2 = (x-1)^2 + 2^2 = 4.\)
Thus, \((x-1)^2 = 0\) and \(x = 1.\)
Since we have a unique solution, condition 1) is sufficient.

Condition 2)
\(1△x = (1-1)^2 + (x+1)^2 = (x+1)^2 = 4.\)
So, \(x+1 = ±2\) or \(x = -1 ± 2.\)
Thus, \(x = -3\) or \(x = 1\).
Since we don’t have 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.
_________________

=>

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.
Visit https://www.mathrevolution.com/gmat/lesson for details.

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.

Since we have \(3\) variables (\(x, y,\) and \(k\)) and \(0\) equations, E is most likely 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 \(x + y = 2\), we have \(y = 2 – x.\)

Substituting \(y = 2 - x\) into \(3x + 5y = k + 1\) gives us \(3x + 5(2 - x) = k + 1, 3x + 10 - 5x = k + 1, -2x + 10 = k + 1\) or \(2x + k = 9.\)

Substituting \(y = 2 - x\) into \(2x + 3y = k\) gives us \(2x + 3(2 - x) = k, 2x + 6 = 3x = k, -x + 6 = k\) or \(x + k = 6.\)

We now have \(2\) equations: \(2x + k = 9\) and \(x + k = 6\). Rewriting the first equation gives us \(k = 9 - 2x\). Substituting this into the second equation gives us \(x + 9 - 2x = 6, -x = -3\), and \(x = 3.\) Then \(x + k = 6\) becomes \(3 + k = 6,\) and \(k = 3.\)

Then we have \(x = 3\) and \(k = 3.\)

Therefore, C is the answer.
Answer: C


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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If |r|>|s|, is r>s?
1) r>0
2) s>0

Is the answer A?

I am not able to see answers after clicking 'spoiler link' in the original post. Is anyone else facing the same issue?

Thanks!

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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)
Assume \(x = 6a\) and \(y = 6b\) where \(a\) and \(b\) are relatively prime.

We have \(x*y = (6a)*(6b)= 1008\) or \(ab = 28\).

Then we have four pairs of \((a,b)\) which are \((1,28), (4,7), (7,4)\) and \((28,1).\)

Thus we have four pairs of \((x,y)\) which are \((6, 168), (24, 42), (42, 24)\) AND \((6,168)\) and we don’t have a unique answer.

Therefore, E is the answer.
Answer: E

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.
_________________

=>

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.
Visit https://www.mathrevolution.com/gmat/lesson for details.

Since we have \(2\) variables (\(p\) and \(q\)) and \(0\) equations, C is most likely 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 \(|a| = 4\), and \(|b| = 3,\) we have \(a = ±4\), and \(b = ±3\) from condition 1) and ab < 0 from condition 2).
Then we have \(a^2 = 16, b^2 = 9 \)and \(ab = -12.\)

Thus \((a - b)^2 = a^2 - 2ab + b^2 = 16 – 2(-12) + 9 = 49.\)

Since both conditions together yield a unique solution, they 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.
_________________

=>

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)
\(\frac{1}{x} + \frac{1}{y} = \frac{1}{5}\)

=> \(5y + 5x = xy\) after multiplying by \(5xy\)

=> \(xy - 5y - 5x = 0\)

=> \(xy - 5y - 5x + 25 = 25\)

=> \((x-5)(y-5) = 25\)

Since \(x\) and \(y\) are positive integers, there are three possible pairs of values for \((x-5,y-5)\). These are are \((1,25)\), \((25,1)\) and \((5,5).\)

So, the possible pairs \((x,y)\) are \((6,30), (30,6)\) and \((10,10).\)

Since we don’t have a unique solution, both conditions together are not sufficient.

Therefore, E is the answer.
Answer: E

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.
_________________

=>

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.
Visit https://www.mathrevolution.com/gmat/lesson for details.

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.

When we simply condition 2), we have
\(<2x, 2y>+1=<y, x>-2\)

\(=> 2x + \frac{2y}{2} + 1 = y + \frac{x}{2} - 2\)

\(=> 2x + y + 1 = y + \frac{x}{2} – 2\)

\(=> (\frac{3}{2})x = -3\)

\(=> x = -2.\)

Condition 2) is sufficient.

Condition 1)
Since we have \(<x,y> = y + \frac{x}{2} = x + \frac{y}{2}\), we have \(\frac{y}{2} = \frac{x}{2}\) or \(x = y.\)

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

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.
Visit https://www.mathrevolution.com/gmat/lesson for details.


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.

Since the mid-point of x and y is 4, (x+y)/2 = 4, and x + y = 8.

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

Condition 1)
Since |x|=6, we must have x=6 or x=-6.
Since 4 lies midway between x and y, we must have either x=6 and y=2, or x=-6 and y=14.
Since condition 1) does not yield a unique solution, it is not sufficient.

Condition 2)
There are many possible pairs of integers, x and y, satisfying condition 2) and the original condition.
Examples are x=1 and y=7, and x=2 and y=6.
Condition 2) is not sufficient since it does not yield a unique solution.

Conditions 1) & 2)
Condition 1) tells us that x=6 and y=2, or x=-6 and y=14.
Since condition 2) tells us that x<y, we must have x=-6 and y=14.
Both conditions together are sufficient, since they yield a unique solution.

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.
_________________

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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)
Both \(n = 6\) and \(n = 10\) are products of two different prime numbers less than \(15\). Thus, condition 1) is not sufficient since it does not yield a unique solution.

Condition 2)
Both \(n = 11\) and \(n = 13\) are relatively prime to \(210\). Thus, condition 2) is not sufficient since it doesn’t yield a unique solution.

Conditions 1) & 2)
The prime numbers less than \(15\) are \(2, 3, 5, 7, 11\) and \(13\).

Since \(n\) and \(210 = 2*3*5*7\) are relatively prime, we must have \(n = 11*13 = 143.\)

Both conditions 1) & 2) together are sufficient since they yield a unique solution.

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.
_________________

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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 \(\frac{x^p}{x^q} = x^{p-q} = x^4 = 2^4 = 16.\)
If \(x = 16, p = 5\) and \(q = 1\), then \(\frac{x^p}{x^q} = x^{p-q} = x^4 = 16^4 = 2^{16} = 65536.\)
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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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 question is equivalent to asking if \(xy < 0\). This can be seen by multiplying both sides of the inequality by \(y^2\).

Since we can ignore even exponents in inequalities like \(x^4y^5<0\), condition 1) is equivalent to the statement \(y < 0\) and condition 2) is equivalent to the statement \(xy < 0\).

Therefore, the 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.

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

The units digits of \(3^k\) have period \(4\) as they form the cycle \(3 -> 9 -> 7 -> 1.\)
\(3^{4m+2}\) has \(9\) as its units digit if \(3^{4m+2}\) has units digit \(9\), regardless of the value of \(m\).
Thus, the divisibility of \(3^{4m+2}+n\) by \(5\) relies on the variable n only.

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 (\(x\)) 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
\(\frac{( 11 + 14 + 16 + 18 + x )}{5} = 14.\)
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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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)
Since \(p + q = r\) and \(p, q\) and \(r\) are prime numbers, one of \(p\) and \(q\) must be \(2.\)

In addition, \(p = 2\) because \(1 < p < q.\)

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)
If \(p = 2, q = 3\) and \(r = 5\), then \(p = 2.\)

If \(p = 3, q = 2\) and \(r = 5\), then \(p = 3.\)

Since condition 1) doesn’t yield a unique solution, it is not sufficient.

Condition 2)
If \(p = 2\) and \(q = 3\), then \(p = 2.\)

If \(p = 3\) and \(q = 5,\) then \(p = 3.\)

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.
_________________

=>

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.
Visit https://www.mathrevolution.com/gmat/lesson for details.

Since we have \(2\) variables (\(x\) and \(y\)) and \(0\) equations, C is most likely 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 \(\frac{1}{x} + \frac{1}{y} = 1\) from condition 2), we have \(y + x = xy\) by multiplying both sides of the equation by \(xy\), which rearranges to get \(xy – (x+y) = 0.\)

Since \(xy + (x+y) = -2\) from condition 1), we have \(xy - (x+y) + xy + (x+y) = 0 + -2\) by adding the two equations. Then \(2xy = -2\) or \(xy = -1\).

Then we have \(x+y=-1.\)

Since both conditions together yield a unique solution, they 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.
_________________

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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)
The factors of \(77\) are \(1, 7, 11\) and \(77\). We have \(n = 7\) since the least factor of \(77\) other than \(1\) is \(7\).
Condition 1) is sufficient.

Condition 2)
\(n\) is a prime number, by definition, since it has exactly two factors.
Condition 2) is sufficient.

Therefore, D is the answer.
Answer: D

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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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) implies that \(m = n = 0\), since \(3^m\) and \(5^n\) are powers of prime numbers that are equal. Thus, condition 1) is sufficient.
Condition 2) also implies that \(m = n = 0\), since \(|m|=- √n\) is only possible when \(m = n = 0\) (\(- √n <=0\) and \(|m| >=0\)). 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).
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