The Ultimate Q51 Guide [Expert Level]

1 + 2 + … + k = k(k+1)/2. m and n are positive integers satisfying n < m. What is the value of n + (n+1) + … + (m-1) + m?

A.(m+n)(m-n-1)/2
B.((m+n)(m-n+1))/2
C.((m-n)(m+n-1))/2
D.((m-n)(m+n+1))/2
E.((m+n)(m-n-1))/4

=>

n + (n+1) + … + (m-1) + m
= 1 + 2 + … + (n-1) + n + (n+1) + … + (m-1) + m – (1 + 2 + … + (n-1))
= m(m+1)/2 - (n-1)(n-1+1)/2
= m(m+1)/2 – n(n-1)/2
= (1/2)(m(m+1) – n(n-1))
= (1/2)(m2 + m – n2 + n)
= (1/2)(m2 – n2 + m + n )
= (1/2)( (m+n)(m-n) + (m+n) )
= (1/2)(m+n)(m-n+1)

Therefore, the answer is B.
Answer: B

If 272/4^6 = 1/4^m + 1/4^n , what is the value of mn?

A. 2 B. 4 C. 6 D. 8 E. 12

=>

272/4^6 = 17/4^4 =(1+16)/4^4 =1/4^4 +16/4^4 =1/4^4 +1/4^2

Thus m = 4 and n = 2, or m = 2 and n = 4.
So, mn = 8.

Therefore, the answer is D.
Answer: D 

[GMAT math practice question]

(inequality) 0 < a < b < c. Is a < 3?

1) 1/c > 1/3
2) 1/a + 1/b + 1/c = 1

=>

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 have 1/a > 1/b > 1/c, since we are given that 0 < a < b < c.

Condition 1) implies that c < 3. Therefore a < 3, and Condition 1) is sufficient.

Condition 2)
Since 1/a > 1/b > 1/c > 0, and 1/a + 1/b + 1/c = 1, we must have 1/a + 1/a + 1/a = 3/a > 1. Therefore, a < 3, and the answer is ‘yes’.
Condition 2) is sufficient since it yields a unique answer.

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]

(inequality) Is x^3-x^2+x-1 > 0?

1) x^5 > x^2
2) x^3 + x > x^2 + 1

=>

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.

In inequality questions, the law “Question is King” tells us that if the solution set of the question includes the solution set of the condition, then the condition is sufficient.

x^3-x^2+x-1 > 0
=> x^2(x-1) + (x-1) > 0
=> (x^2+1)(x-1) > 0
=> (x-1) > 0, since x^2+1 > 0
=> x>1
The question asks if x > 1.

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^5 > x^2
=> x^5 - x^2 > 0
=> x^2(x^3 – 1) > 0
=> x^2(x-1)(x^2+x+1) > 0
=> x – 1 > 0 since x^2+x+1 >0, and x^2 > 0 if x ≠ 0
x > 1
This condition is equivalent to the question. Therefore, condition 1) is sufficient.

Condition 2)
x^3 + x > x^2 + 1
=> x^3 - x^2 + x – 1 > 0
=> x^2(x-1)+ (x-1) > 0
=> (x^2+1)(x-1) > 0
=> (x-1) > 0, since x2+1 > 0
=> x > 1
This condition is equivalent to the question. Therefore, 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).

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.

[GMAT math practice question]

(number properties) If p and q are prime numbers, what is the number of the different factors of p^2q^3?

1) pq=143
2) p and q are different

=>

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 p and q are prime numbers, condition 1) tells us that p = 11 and q = 13, or p = 13 and q = 11. Therefore, since p and q are different prime numbers, the number of different factors of p^2q^3 is (2+1)(3+1) = 12. Condition 1) is sufficient since it yields a unique solution.


Condition 2)
Since condition 2) tells us that p and q are different prime numbers, the number of factors of p^2q^3 is (2+1)(3+1) = 12.
Condition 2) is sufficient since it yields 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).

[GMAT math practice question]

(inequality) Is x<0?

1) x^3+1<0
2) x^3+2x^2+x+2=0

=>

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)
x^3+1<0
⇔ (x+1)(x^2-x+1) < 0
⇔ x+1<0 since x^2-x+1 > 0
⇔ x < -1
In inequality questions, the law “Question is King” tells us that if the solution set of the question includes the solution set of the condition, then the condition is sufficient
Thus, since the solution set of the question, ‘x<0’, includes that of condition 1), ‘x<-1’, condition 1) is sufficient.

Condition 2)
x^3+2x^2+x+2=0
⇔ x^2(x+2)+(x+2)=0
⇔ (x^2+1)(x+2)=0
⇔ x+2=0 since x^2+1 > 0
⇔ x = -2
Thus, x < 0 and 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.

[GMAT math practice question]

(geometry) a, b and c are positive numbers. Is a>b-c?

1) a, b, and c are the lengths of three different sides of a triangle
2) a^2+b^2=c^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. 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

=>

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]

(equation) What is the value of the following sum?

1/(1*2) + 1/(2*3) + 1/(3*4) + 1/(4*5) + 1/(5*6)

A. 2/3
B. 3/4
C. 4/5
D. 5/6
E. 1

=>

Recall that 1/[n*(n+1)] = 1/n – 1/(n+1).

So,
1/(1*2) + 1/(2*3) + 1/(3*4) + 1/(4*5) + 1/(5*6)
= (1/1 – 1/2) + (1/2 – 1/3) + (1/3 – 1/4) + (1/4 – 1/5) + (1/5 – 1/6)
= 1/1 – 1/6 = 5/6

Therefore, the answer is D.
Answer: D

[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

=>

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

=>

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]

(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) m / n is an integer.

=>

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

[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.

=>

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 ( 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) What is the average (arithmetic mean) of all 5-digit numbers that can be formed using each of the digits 1, 3, 5, 7 and 9 exactly once?

A. 48000
B. 50000
C. 54000
D. 55555
E. 56000

=>

There are 5! = 120 different numbers of this form.


Next, we find the sum of all numbers of this form.

Let’s start by considering the numbers with 1 in each position. There are 4! = 24 numbers with 1 in the ten-thousands digit (these contribute 24 x 1 x 10000 to the sum), 24 numbers with 1 in the thousands digit (these contribute 24 x 1 x 1000 to the sum) 24 numbers with 1 in the hundreds digit (these contribute 24 x 1 x 100 to the sum), 24 numbers with 1 in the tens digit (these contribute 24 x 1 x 10 to the sum), and 24 numbers with 1 in the units digit (these contribute 24 x 1 x 1 to the sum).

Thus, the digit 1 contributes a total of 24 x 1 x (10000 + 1000 + 100 + 10 + 1) = 24 x 1 x 11111 to the sum. Similarly, the 3s contribute 24 x 3 x 11111 to the sum, and so on. Thus, the sum of all numbers of this form is
24 * ( 1 + 3 + 5 + 7 + 9 ) * 11111 = 24*25*11111 = 66666600.

The average (arithmetic mean) of these numbers is 66666600 / 120 = 555555.

Therefore, the answer is D.
Answer: D

[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

=>

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 k = 2, m = 3 and n = 5, then kmn = 30.
If k = 1, m = 3 and n = 80, then kmn = 240.

Conditions 1) & 2) together are not sufficient, when applied together, since they don’t yield a unique solution.

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]

(statistics) k, m and n are positive integers. Is their average equal to their median?

1) The median of k, m and n is 11.
2) The range of k, m and n is 13

=>

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.

Without loss of generality, we may assume k ≤ m ≤ n.
If their average and their median are equal, then ( k + m + n ) / 3 = m or k + m + n = 3m, so n + k = 2m and n-k=2m-2k. So, the range of the numbers must be even.
Thus, condition 2) yields the unique answer ‘no’, and is sufficient by CMT 1).

Condition 1)
If k = 10, m = 11 and n = 12, then their average and median are equal, and the answer is ‘yes’.
If k = 10, m = 11 and n = 13, then their average and their median are not equal, 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

[GMAT math practice question]

(absolute value) Is x > 1?

1) x^2 > x|x|
2) x|x| > x

=>

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)
x^2 > x|x|
=> |x|^2 > x|x|
=> |x|^2 - x|x| > 0
=> |x|(|x| - x) > 0
=> (|x| - x) > 0
=> |x| > x
=> x < 0
Thus, condition 1) is sufficient since it yields the unique answer, ‘no’.

Condition 2)
x|x| > x
Assume x ≥ 0. x|x| > x ⇔ x^2 > x ⇔ x^2 – x > 0 ⇔ x(x-1) > 0 ⇔ x < 0 or x > 1.
We must have x > 1 by the assumption.

Assume x < 0. x|x| > x ⇔ -x^2 > x ⇔ x^2 + x < 0 ⇔ x(x+1) < 0 ⇔ -1 < x < 0.
In this case, -1 < 0 < 1.

So, condition 2) tells us that -1 < x < 0 or x > 1.

In inequality questions, the law “Question is King” tells us that if the solution set of the question includes the solution set of the condition, then the condition is sufficient

The solution set of the question “x > 1” doesn’t include the solution set of the condition 2) “-1 < x < 0 or x > 1”. Condition 2) is not sufficient.

Therefore, A is the answer.
Answer: A