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Math Revolution DS Expert - Ask Me Anything about GMAT DS 22 Aug 2019, 00:58
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[GMAT math practice question]

(algebra) What is the value of the integer \(a\)?

1) \(x - (\frac{2}{3})(x-4a) = 7\) has a positive integer solution

2) \(a\) is positive
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Math Revolution DS Expert - Ask Me Anything about GMAT DS Updated on: 13 Mar 2022, 04:10
Originally posted by MathRevolution on 23 Aug 2019, 01:43.
Last edited by MathRevolution on 13 Mar 2022, 04:10, edited 1 time in total.
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[GMAT math practice question]

(set) \(A={x|(\frac{7}{15})x+\frac{1}{3} = \frac{4}{3}}\) and \(B={y| 2m-(\frac{1}{15})y = 3}\), where \(m\) is a real number. What is the value of \(m\)?

\(1) A∩B≠Ø\)

\(2) B≠Ø\)
Show more

=>

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.

Since \((\frac{7}{15})x+\frac{1}{3} = \frac{4}{3}\) and \(7x + 5 = 20\) by definition of set \(A\), we must have \(x = \frac{15}{7}\) and \(A = { \frac{15}{7} }.\)

Since \(2m-(\frac{1}{15})x = 3\) and \(30m – x = 45\) by definition of set \(B\), \(x = 30m – 45\) and \(B = { 30m – 45 }.\)

Since we have \(1\) variable (\(m\)) and \(0\) equations, D is most likely to be the answer. So, we should consider each condition on its own first.

Condition 1)
Since \(A∩B≠Ø\), we must have \(30m – 45 = \frac{15}{7}\) and condition 1) yields a unique solution. It is sufficient.

Condition 2)
Since m can be any value and condition 2) doesn’t yield a unique solution, it 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.
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Math Revolution DS Expert - Ask Me Anything about GMAT DS 23 Aug 2019, 01:44
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[GMAT math practice question]

(number properties) Given two different positive integers, what is the ratio of the larger number to the smaller one?

1) the sum of the two numbers is 1000 less than the product of the two numbers
2) one of the two numbers is a perfect square
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Math Revolution DS Expert - Ask Me Anything about GMAT DS Updated on: 13 Mar 2022, 04:11
Originally posted by MathRevolution on 25 Aug 2019, 18:48.
Last edited by MathRevolution on 13 Mar 2022, 04:11, edited 1 time in total.
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[GMAT math practice question]

(algebra) What is the value of the integer \(a\)?

1) \(x - (\frac{2}{3})(x-4a) = 7\) has a positive integer solution

2) \(a\) is positive
Show more

=>


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 \(a\)) 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)
\(x - (\frac{2}{3})(x - 4a) = 7\) is equivalent to \(3x – 2(x-4a) = 21\) or \(x = 21 – 8a.\)

The possible pairs \((x,a)\) are \((13,1)\) and \((5,2).\)

Since both conditions together don’t yield a unique solution, they 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.
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Math Revolution DS Expert - Ask Me Anything about GMAT DS 25 Aug 2019, 18:50
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[GMAT math practice question]

(number properties) Given two different positive integers, what is the ratio of the larger number to the smaller one?

1) the sum of the two numbers is 1000 less than the product of the two numbers
2) one of the two numbers is a perfect square
Show more

=>

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 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) and 2)
Suppose \(x\) and \(y\) are the integers and \(x\) is a perfect square.

Then \(xy = x + y + 1000\), and \(xy – x – y + 1 = 1001.\)

Thus, \((x-1)(y-1) = 1001 = 7*11*13.\)

Since \(x\) is a perfect square, only \(11*13 + 1 = 144\) is a perfect square out of all possible values \(7+1, 11+1, 13+1, 7*11+1, 7*13+1, 11*13+1, and 7*11*13+1.\)

Thus, \(x = 144\) and \(y = 8.\)

Therefore, \(x : y = 144:8.\)

Since both conditions together yield a unique solution, they 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 \(xy = x + y + 1000\) or \(xy – x – y + 1 = 1001.\)

Thus \((x-1)(y-1) = 1001 = 7*11*13.\)

We can find pairs of solutions \(x=2\) and \(y=1002\), and \(x=1002\) and \(y=2.\)

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

Condition 2)
Since it provides no information about the second number, condition 2) is not sufficient.

Therefore, the answer is C.
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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Math Revolution DS Expert - Ask Me Anything about GMAT DS 26 Aug 2019, 00:35
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[GMAT math practice question]

(number properties) What is the remainder when \(9^n -1\) is divided by \(10\)?

1) \(n\) is a multiple of \(2\)

2) \(n\) is a multiple of \(3\)
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Math Revolution DS Expert - Ask Me Anything about GMAT DS 27 Aug 2019, 02:21
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[GMAT math practice question]

(number properties) \(A\) and \(B\) are positive integers. \(G\) is the greatest common divisor of \(A\) and \(B\), and \(L\) is the least common multiple of \(A\) and \(B\). What is the value of \(A+B\)?

\(1) \frac{G}{A} + \frac{G}{B} = \frac{7}{10}\)

\(2) L=70\)
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Math Revolution DS Expert - Ask Me Anything about GMAT DS Updated on: 04 Apr 2022, 02:52
Originally posted by MathRevolution on 28 Aug 2019, 00:52.
Last edited by MathRevolution on 04 Apr 2022, 02:52, edited 1 time in total.
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[GMAT math practice question]

(number properties) What is the remainder when \(9^n -1\) is divided by \(10\)?

1) \(n\) is a multiple of \(2\)

2) \(n\) is a multiple of \(3\)
Show more

=>

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.

The remainder when \(9^n -1\) is divided by \(10\) is the same as the units digit of \(9^n -1.\) This is easily determined from the units digit of \(9^n.\)

\(9^1 = 9, 9^2 = 81 ~ 1, 9^3 ~ 9, 9^4 ~ 81 ~ 1, …\)

So, the units digits of 9n have period \(2\):

They form the cycle \(9 -> 1.\)

Thus, \(9^n\) has a units digit of \(9\), if \(n\) is an odd number and a units digit if \(1\), if \(n\) is an even number.

Thus, condition 1) is sufficient.

Condition 2) is not sufficient since a multiple of \(3\) can be either an even or an odd number.

Therefore, A is the answer.
Answer: A
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Math Revolution DS Expert - Ask Me Anything about GMAT DS 28 Aug 2019, 00:52
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[GMAT math practice question]

(number properties) \(N\) is a positive integer. What is the value of \(N\)?

1) \(N\) is divisible by \(75\)

2) \(N\) has \(75\) positive factors, including \(1\) and \(N.\)
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Math Revolution DS Expert - Ask Me Anything about GMAT DS Updated on: 04 Apr 2022, 02:53
Originally posted by MathRevolution on 29 Aug 2019, 23:43.
Last edited by MathRevolution on 04 Apr 2022, 02:53, edited 1 time in total.
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[GMAT math practice question]

(number properties) \(A\) and \(B\) are positive integers. \(G\) is the greatest common divisor of \(A\) and \(B\), and \(L\) is the least common multiple of \(A\) and \(B\). What is the value of \(A+B\)?

\(1) \frac{G}{A} + \frac{G}{B} = \frac{7}{10}\)

\(2) L=70\)
Show more

=>

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 (\(A\) and \(B\)) 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)
Suppose \(A=aG\) and \(B=bG\) for some integers \(a, b\) and \(G,\) where \(a\) and \(b\) are relatively prime.

Then
\(\frac{G}{A} + \frac{G}{B} = \frac{7}{10}\)
\(=> \frac{G}{(aG)} + \frac{G}{(bG)} = \frac{7}{10}\), since \(A=aG\) and \(B = bG\)
\(=> \frac{bG}{(abG)} + \frac{aG}{(abG)} = \frac{7}{10}\), taking a common denominator
\(=> \frac{(aG+bG)}{(abG)} = \frac{7}{10}\)
\(=> \frac{(A+B)}{L} = \frac{7}{10}\)
\(=> \frac{(A+B)}{70} = \frac{7}{10}\), since \(L = 70\)
\(=> A+B = 49\)
Since both conditions together yield a unique solution, they 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 A = 2 and B = 5, then A + B = 7.
If A = 6 and B = 15, then A + B = 21.
Since condition 1) doesn’t yield a unique solution, it is not sufficient.

Condition 2)
If A = 14 and B = 35, then A + B = 49.
If A = 2 and B = 35, then A + B = 37.
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.
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Math Revolution DS Expert - Ask Me Anything about GMAT DS 29 Aug 2019, 23:44
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[GMAT math practice question]

(Number Properties) \(gcd(x,y)\) is defined to be the greatest common divisor of \(a\) and \(b\). What is the greatest common divisor of \(a, b, c\)?

\(1) gcd(a, b)= 18\)

\(2) gcd(b, c)= 24\)
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Math Revolution DS Expert - Ask Me Anything about GMAT DS Updated on: 24 May 2021, 03:49
Originally posted by MathRevolution on 01 Sep 2019, 22:52.
Last edited by MathRevolution on 24 May 2021, 03:49, edited 1 time in total.
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[GMAT math practice question]

(number properties) \(N\) is a positive integer. What is the value of \(N\)?

1) \(N\) is divisible by \(75\)

2) \(N\) has \(75\) positive factors, including \(1\) and \(N.\)
Show more

=>

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 \(1\) 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 possible values of \(N\) are \(75, 150,\) …. Since condition 1) does not yield a unique solution, it is not sufficient.

Condition 2)
\(2^{74}\) and \(2^4*3^4*5^2\) have \(75\) factors. Since condition 2) does not yield a unique solution, it is not sufficient.

Conditions 1) & 2)
\(N = 3^{24}*5^2\) and \(N=2^4*3^4*5^2\) have \(75\) factors. Since both conditions together do not yield a unique solution, they are not sufficient.

Therefore, E is the answer.
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.
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Math Revolution DS Expert - Ask Me Anything about GMAT DS Updated on: 24 May 2021, 03:50
Originally posted by MathRevolution on 01 Sep 2019, 22:54.
Last edited by MathRevolution on 24 May 2021, 03:50, edited 1 time in total.
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[GMAT math practice question]

(Number Properties) \(gcd(x,y)\) is defined to be the greatest common divisor of \(a\) and \(b\). What is the greatest common divisor of \(a, b, c\)?

\(1) gcd(a, b)= 18\)

\(2) gcd(b, c)= 24\)
Show more

=>

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 \(3\) variables (\(a, b\) and \(c\)) 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 \(gcd(a,b) = 18 = 2^1*3^2\) and \(gcd(b,c) = 24 =2^3*3^1\),\(gcd(a,b,c) = gcd(18,24) = gcd( 2^1*3^2 , 2^3*3^1 ) = 2^1*3^1 = 6\) by choosing the minimum exponents from the prime factorizations. 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)
Since it provides no information about the value of c, condition 1) is not sufficient.

Condition 2)
Since it provides no information about the value of a, condition 2) is 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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Math Revolution DS Expert - Ask Me Anything about GMAT DS 01 Sep 2019, 22:54
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[GMAT math practice question]

(Absolute Values) What is the value of \(x+y\)?

\(1) |x-2|=4\)

\(2) |x-y+3|=4\)
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Math Revolution DS Expert - Ask Me Anything about GMAT DS Updated on: 19 Nov 2021, 07:47
Originally posted by MathRevolution on 01 Sep 2019, 22:56.
Last edited by MathRevolution on 19 Nov 2021, 07:47, edited 1 time in total.
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[GMAT math practice question]

(Absolute Values) What is the value of \(x+y\)?

\(1) |x-2|=4\)

\(2) |x-y+3|=4\)
Show more

=>

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 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 \(|x-2| = 4\), we have \(x – 2 = ±4.\) So, \(x = 2 ± 4.\)

Thus, \(x = 6\) or \(x = -2\)

Since \(|x – y + 3| = 4,\) we have \(x – y + 3 = ±4\) and \(y = x + 3 ± 4.\)

Thus, \(y = x – 1\) or \(y = x + 7.\)

The possible pairs \((x,y)\) are \((6,5), (6,13), (-2,-3)\) and \((-2,5),\) and their sums are \(11, 19, -5\) and \(3.\)

Since both conditions together do not yield a unique solution, they are not sufficient.

Therefore, the answer is E.
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.
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Math Revolution DS Expert - Ask Me Anything about GMAT DS 02 Sep 2019, 01:23
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[GMAT math practice question]

(number properties) What is the value of \(x\)?

1) the remainder, when \(170\) is divided by \(x\), is \(2\)

2) the remainder, when \(140\) is divided by \(x\), is \(4\)
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Math Revolution DS Expert - Ask Me Anything about GMAT DS 03 Sep 2019, 00:38
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[GMAT math practice question]

(number properties) For positive integers \(a, b, c, d\) and \(e\), what is the tuple \((a, b, c, d, e)\)?

\(1) ab=72, bc=108\)

\(2) cd=60, de=500\)
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Math Revolution DS Expert - Ask Me Anything about GMAT DS Updated on: 19 Nov 2021, 07:48
Originally posted by MathRevolution on 04 Sep 2019, 00:44.
Last edited by MathRevolution on 19 Nov 2021, 07:48, edited 1 time in total.
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[GMAT math practice question]

(number properties) What is the value of \(x\)?

1) the remainder, when \(170\) is divided by \(x\), is \(2\)

2) the remainder, when \(140\) is divided by \(x\), is \(4\)
Show more

=>

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 \(1\) 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)
Now, \(170 = x*a + 2,\) so \(168 = 2^3*3*7 = x*a\). Note that \(x > 2\) since the dividend must be greater than the remainder.

So, \(x\) is a factor of \(168\) greater than \(2\). The possible values of \(x\) are \(3, 4, 6, …, 168.\)

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

Condition 2)
Now, \(140 = x*b + 4\), so \(136 = x*b\). Note that \(x > 4\) since the dividend must be greater than the remainder.

So, \(x\) is a factor of \(136 = 2^3*17\) greater than \(4\). The possible values of \(x\) are \(8, 17\) and \(140.\)

Since condition 2) doesn’t yield a unique solution, it is not sufficient.

Conditions 1) & 2).
When we consider both conditions together, \(x\) is a common factor greater than \(4\) of \(168 = 2^3*3*7\) and \(136 = 2^3*17\). The only possible value of \(x\) is \(8.\)

Since both conditions together yield a unique solution, they 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.
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Math Revolution DS Expert - Ask Me Anything about GMAT DS 04 Sep 2019, 00:45
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[GMAT math practice question]

(number properties) What is the value of \(x\)?

1) the prime factorization of \(x\) is \(ab(10a+b)\) (\(a, b\) are positive integers less than or equal to \(9\))

2) \(x\) is \(4\)-digit number
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Math Revolution DS Expert - Ask Me Anything about GMAT DS 05 Sep 2019, 01:15
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[GMAT math practice question]

(number properties) For positive integers \(a, b, c, d\) and \(e\), what is the tuple \((a, b, c, d, e)\)?

\(1) ab=72, bc=108\)

\(2) cd=60, de=500\)
Show more

=>

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 \(5\) variables (\(a, b, c, d\) and \(e\)) 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 \(ab = 2^3*3^2, bc = 2^2*3^3, cd = 2^2*3*5\) and \(de = 2^2*5^3\), \(b\) is a common factor of \(2^3*3^2\) and \(2^2*3^3\), \(c\) is a common factor of \(2^2*3^3\) and \(2^2*3*5\), and \(d\) is a common factor of \(2^2*3*5\) and \(2^2*5^3.\)

So, \(b\) is a factor of \(2^2*3^2\), \(c\) is a factor of \(2^2*3\) and \(d\) is a factor of \(2^2*5.\)
Two possible solutions are \(a=2, b=2^2*3^2, c=3, d=2^2*5\) and \(e=5^2\), and \(a=4, b=2*3^2, c=2*3, d=2*5\) and \(e=2*5^2.\)
Since both conditions together do not yield a unique solution, 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.
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