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[Math Revolution GMAT math practice question]

(number properties) If \(x\) and \(y\) are positive integers and \(y=\sqrt{5-x}\), then \(y\)=?

\(1) x>1\)
\(2) y<2\)

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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.

[\(x\)] is analyzed as follows.
If \(n ≤ x < n + 1\) for some integer n, then \([x] = n\).

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)
\([x] = 2\)
\(=> 2 ≤ x < 3\)
Thus, condition 1) is not sufficient, since it does not yield a unique solution.

Condition 2)
Since there are a lot of integers, condition 2) does not yield a unique solution. This condition is not sufficient.

Conditions 1) & 2)
\(x = 2\) is the unique integer such that \(2 ≤ x < 3.\)
Thus, 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]

(number property) If \(n\) is a positive integer, is \(\sqrt{n+1}\) an integer?

1) \(n\) is a multiple of \(8\)
2) \(n\) is the product of \(2\) consecutive even numbers

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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.

Modifying the original condition:
Note that for \(\sqrt{5-x}\) to make sense, we must have \(x ≤ 5.\)
Also,
\(y=\sqrt{5-x}\),
\(=> y^2 = 5 – x.\)
Since \(y^2 = 5 – x\) is the square of an integer and \(0< x ≤ 5\), the only possible solutions are \(x = 1, y = 2\) and \(x = 4, y = 1.\)
Thus, if a condition allows us to figure out the value of either \(x\) or \(y\), it is sufficient.

Condition 1)
Since \(x > 1\), we must have \(x = 4\) and \(y = \sqrt{5-x} =\sqrt{5-4} = 1.\)
Thus, condition 1) is sufficient since it gives a unique solution.

Condition 2)
Since \(y < 2\), we have \(y = 1\) from the original condition.
Thus, condition 2) is sufficient since it gives a unique solution.

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).

[Math Revolution GMAT math practice question]

(number properties) If \(m\) and \(n\) are positive integers, is \(mn\) an even number?

1) \(\frac{m}{n}\) is an even number.
2) \(m + n\) is an even number.

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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.

Condition 1)
If \(n = 8\), then \(\sqrt{n+1} = \sqrt{8+1} = \sqrt{9} = 3\) and the answer is ‘yes’.
If \(n = 16\), then \(\sqrt{n+1} = \sqrt{16+1} = \sqrt{17}\) and the answer is ‘no’.
Thus, condition 1) is not sufficient since it does not yield a unique solution.

Condition 2)
If \(n\) is the product of two consecutive even integers, then
\(n = 2k(2k+2) = 4k^2 + 4k\) for some integer \(k\).
\(\sqrt{n+1} = \sqrt{4k^2+4k+1} = \sqrt{(2k+1)^2} = 2k+1,\) and the answer is ‘yes’.
Thus, 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) When \(n\) is a positive integer, is \(\frac{n}{4}\) an integer?

1) \(n - 1\) is not divisible by \(2\)
2) \(n + 1\) is not divisible by \(2\)

=>

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.

Modifying the question:
\(mn\) is an even number precisely when at least one of \(m\) and \(n\) is even. So,
the question asks if either \(m\) or \(n\) is even.

Condition 1:
If \(\frac{m}{n} = 2k\) for some integer \(k\), then \(m = 2kn\), which is an even number.
Thus, condition 1) is sufficient.

Condition 2)
If \(m = 2\) and \(n = 4\), then \(m + n = 6\) is even, and \(mn = 8\) is an even number, so the answer is ‘yes’.
If \(m = 1\) and \(n = 3\), then \(m + n = 4\) is even, and \(mn = 3\) is not an even number, so the answer is ‘no’.
Since it does not give us a unique answer, condition 2) is not sufficient.

Therefore, the correct answer is A.
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.

Modifying the question:
Asking whether \(\frac{n}{4}\) is an integer is equivalent to asking whether \(m\) is a multiple of \(4\).

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)
Since \(n – 1\) is not divisible by \(2, n – 1\) is an odd number and \(n\) is an even number.
If \(n = 4\), then \(n\) is a multiple of \(4\) and the answer is ‘yes’.
If \(n = 2,\) then \(n\) is not a multiple of \(4\) and the answer is ‘no’.
Thus, condition 1) is not sufficient, since it does not yield a unique solution.

Condition 2)
Since \(n + 1\) is not divisible by \(2, n + 1\) is an odd number and \(n\) is an even number.
If \(n = 4\), then \(n\) is a multiple of \(4\) and the answer is ‘yes’.
If \(n = 2\), then \(n\) is not a multiple of \(4\) and the answer is ‘no’.
Thus, condition 2) is not sufficient, since it does not yield a unique solution.

Conditions 1) & 2):
If \(n = 4,\) then neither \(n – 1\) nor \(n + 1\) is divisible by \(2\), but \(n\) is a multiple of \(4\) and the answer is ‘yes’.
If \(n = 2\), then neither \(n – 1\) nor \(n + 1\) is divisible by \(2\), and \(n\) is not a multiple of \(4\) and the answer is ‘no’.
Thus, both conditions together are not sufficient, since they do not yield a unique solution.

Therefore, the answer is E.
Answer: E

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) \([x]\) is the greatest integer less than or equal to \(x\). \(<x>\) is the least integer greater than or equal to \(x\). What is the value of \(x\)?

\(1) [x] = 2\)
\(2) <x> = 2\)

[Math Revolution GMAT math practice question]

(number properties) \(x\) and \(y\) are positive integers. Is \(y\) an even integer?

\(1) x^2+x=y+2\)
\(2) x = 2\)

=>

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.

\([x]\) is analyzed as follows.
If \(n ≤ x < n + 1\) for some integer \(n\), then \([x] = n.\)
\(<x>\) is analyzed as follows.
If \(n – 1 < x ≤ n\) for some integer \(n\), then \(<x> = n.\)

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)
\([x] = 2\)
\(=> 2 ≤ x < 3\)
Thus, condition 1) is not sufficient, since it does not yield a unique solution.

Condition 2)
\(<x> = 2\)
\(=> 1 < x ≤ 2\)
Thus, condition 2) is not sufficient, since it does not yield a unique solution.

Conditions 1) & 2)
Only \(x = 2\) satisfies both conditions.
Since the answer is unique, 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 \(\frac{x}{y}>1\)?

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

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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 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 \(y = x^2 + x – 2\) and \(x = 2\), we have \(y = 4\).
Since this answer is unique, 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)
Since \(y = x^2 + x – 2 = (x-1)(x+2)\) and \(x\) is an integer, one of \(x – 1\) and \(x + 2\) is an even integer.
Thus, \(y\) is always an even integer and condition 1) is sufficient.

Condition 2)
Since it provides no information about y, condition 2) is not sufficient.


Therefore, A is the answer.
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.

[Math Revolution GMAT math practice question]

(absolute value) Is \(|m-n|=|m|-|n|\) ?

\(1) m-n = 0\)
\(2) n = 0\)

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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.

\(\frac{x}{y}>1\)
\(=> xy>y^2\)
\(=> xy-y^2>0\)
\(=> y(x-y)>0\)
By condition 2), \(x-y > 1 > 0\), but we can’t determine whether \(y\) is positive from condition 1).

Therefore, E is the answer.
Answer: E

[Math Revolution GMAT math practice question]

(integer) \(n\) is a positive integer. Is \(\frac{n(n+1)(n+2)}{4}\) an even integer?

1) \(n\) is an even integer
2) \(1238 ≤ n ≤ 1240\)