The Ultimate Q51 Guide [Expert Level]

[GMAT math practice question]

(Algebra) What is the value of x^5 + y^5?

1) x + y = 3.
2) xy = 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.
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 (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)

We have x = 1, y = 2 or x = 2, y = 1.
Then we have x^5 + y^5 = 33 for both cases.
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.

[GMAT math practice question]

(Algebra) x and y are positive integers satisfying 0 < x - √3y < 1 and b is the decimal portion of (x + √3y)3. What is the value of (x - √3y)3 in terms of b?

A. b
B. -b
C. 1 + b
D. 1 - b
E. 3b

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Assume a is the decimal portion of (x - √3y)^3.
Since the integer portion of (x - √3y)^3 is 0, we have a = (x - √3y)^3.
If c is the integer portion of (x + √3y)^3, we have (x + √3y)^3 = c + b.
Then, we have (x - √3y)^3 + (x + √3y)^3 = 2(x^3 + 9xy^2) = a + c + b.
a + b = 2(x^3 + 9xy^2) - c
Since x, y, and c are positive integers, a + b is an integer.
Since we have 0 < a < 1 and 0 ≤ b < 1, we have 0 < a + b < 2.
Thus a + b = 1.
(x - √3y)^3 = a = 1 - b.

Therefore, D is the answer.
Answer: D

[GMAT math practice question]

(Algebra) x, y, and z are real numbers. What is the value of y/x?

1) x and y are positive numbers.
2) x/y=2y/x-z=2x+y/z.

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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 we have 3 variables (x, y, and z) 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)

We have x^2 - zx = 2y^2 when we cross multiply x/y=2y/x-z.

We have 2xy + y^2 = zx when we cross multiply x/y=2x+y/z.

When we subtract the second equation from the first equation, we have
x^2 – zx – (2xy + y^2) = 2y^2 – zx
x^2 – zx – 2xy – y^2 = 2y^2 – zx (multiplying -1 through the bracket)
x^2 – zx – 2xy – y^2 – 2y^2 + zx = 0 (moving all terms to the left side of the equation)
x^2 - 2xy - 3y^2 = 0 (gathering like terms)
(x - 3y)(x + y) = 0 (factoring)
Since x > 0 and y > 0 from condition 1), we have x + y ≠ 0 and x = 3y.
Thus y/x = y/3y = 1/3.

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

[GMAT math practice question]

(Number Properties) x in an integer. Is x a perfect square?

1) x is one greater than the product of 4 consecutive positive integers.
2) x is the sum of five consecutive odd 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.
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 1 variable (x) and 0 equations, D is most likely the answer. So, we should consider each condition on its own first.

Condition 1)

Assume x = k(k + 1)(k + 2)(k + 3) + 1.
Then x = (k^2 + 3k)(k^2 + 3k + 2) + 1 = A(A + 2) + 1 = A^2 + 2A + 1 = (A + 1)2 for A = k^2 + 3k for an integer k.

Thus, x is a perfect integer, and the answer is ‘yes’.

Since condition 1) yields a unique solution, it is sufficient.

Condition 2)

If x = 1 + 3 + 5 + 7 + 9 = 25, then x is a perfect integer and the answer is ‘yes’.
If x = 3 + 5 + 7 + 9 + 11 = 35, then x is not a perfect integer and the answer is ‘no’.

Since condition 2) does not 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.

[GMAT math practice question]

(Number Properties) Is 111…11 – 222…22 a perfect square, where 111…11 and 222…22 are n and m digit numbers, respectively?


1) n = 2m.
2) m = 5.

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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.

111…11 – 222…22 = 999…99 / 9 – 2(999…99/9) = (10^n - 1)/9 – 2(10^m - 1) / 9
= [(10^n - 1) - 2(10^m - 1)] / 9
= [(10^n – 1 + 2*10^m + 2)] / 9
= [(10^2m – 2*10^m + 1)] / 9
= (10^m – 1)^2 / 9
= [(10^m – 1) / 3]^2, if n = 2m from condition 1).

Since condition 1) yields a unique solution, it is sufficient.

Condition 2)

Since we don’t have any information about n, condition 2) does not yield a unique solution, and it is not sufficient.

Therefore, A is the answer.
Answer: A

Since both conditions together are trivial, C is not an answer. If one condition includes a ratio and the other condition just gives a number, the condition, including the ratio is most likely to be sufficient by Tip 4. This tells us that A is most likely to be the answer to this question.

[GMAT math practice question]

(Number Properties) x^3 + y^3 = A(x + y), x^2 + y^2 = B, where x and y are positive integers. What is the value of xy?
1) A - B = 36.
2) B = 97.

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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.

We can simplify the equation x^3 + y^3 = A(x + y) found in the original equation as follows:
x^3 + y^3 = A(x + y)
=> (x + y)(x^2 – xy + y^2) = A(x + y) (factoring using the sum of cubes)
=> [(x + y)(x^2 – xy + y^2)] / (x + y) = A (dividing both sides by (x + y))
=> (x^2 – xy + y^2) = A (simplifying)

Since xy = x^2+xy+y^2 – (x^2+y^2) = (x^3+y^3)/(x+y) – (x^2+y^2) = A – B = 36, condition 1) is sufficient.

Condition 2)
Since x^2 + y^2 = 97, the possible pairs of (x, y) are (4, 9) and (9, 4).
If x = 4, y = 9, then we have xy = 36.
If x = 9, y = 4, then we have xy = 36.

Since condition 2) yields a unique solution, it is sufficient.

Therefore, D is the answer.
Answer: D

This question is a CMT 4(B) question: condition 1) is easy to work with, and condition 2) is difficult to work with. For CMT 4(B) questions, D is most likely the answer, since this question is an integer question.

Sir I am getting value of XY as -36 through first statement.
If you can explain whether negative sign is not considered in here or where I am lacking with in your given steps. That will be grateful. Thank You.

Posted from my mobile device

I have not gone through the question and explanation but could realise from your question and the question stem that you are missing on the important information.
X and Y are positive integers so XY>0

MathRevolution, chetan2u
I am also getting value of XY as -36 through first statement, so there is problem in question statement. XY cannot be positive.

Hi ammuseeru and yashikaaggarwal
Yes the statement 1 is faulty.
What is \(x^3+y^3=(x+y)(x^2+y^2-xy)=(x+y)A........A=(x^2+y^2-xy)=B-xy\)
So A <B and A-B cannot be 36, A-B=-36

[GMAT math practice question]

(Algebra) a = b + c = d + e + f. What is the expression of a(ad + be + df) + be(c + f) + (d + e)f^2 + ce(c + f) + f^3?

A. a
B. a^2
C. a^3
D. a^4
E. a^5


=>

a(ad + be + df) + be(c + f) + (d + e)f^2 + ce(c + f) + f^3
= a(ad + be + df) + be(c + f) + (d + e + f)f^2 + ce(c + f) (combining the terms (d + 3)f^2 and f^3)
= a(ad + be + df) + (b + c)e(c + f) + (d + e + f)f^2 (taking out a common factor of e(c + f) from be(c + f) and ce(c + f))
= a(ad + be + df) + ae(c + f) + af^2 (since a = b + c = d + e + f)
= a(ad + be + df] + a(ce + ef + f^2)
= a[ad + be + df + ce + ef + f^2]
= a[ad + (b + c)e + (d + e + f)f]
= a[ad + ae + af], since a = b + c = d + e + f
= a^2(d + e + f)
= a^3, since a = d + e + f

Therefore, C is the answer.
Answer: C

[GMAT math practice question]

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

1) |x - 2| = 4.
2) |x – y + 3| = 4.

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Forget conventional ways of solving math questions. For DS problems, the VA (Variable Approach) method is the quickest and easiest way to find the answer without actually solving the problem. Remember that equal numbers of variables and independent equations ensure a solution.
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)

We have x = 6 or x = -2, since |x - 2| = 4 ⇔ x - 2 = ±4 ⇔ x = 2 ± 4 ⇔ x = 6 or x = -2 from condition 1).
We have x – y = 1 or x – y = -7, since |x – y + 3| = 4 ⇔ x – y + 3 = ±4 ⇔ x - y = -3 ± 4 ⇔ x - y = 1 or x – y = -7 from condition 2).
If x = 6 and x – y = 1, then we have x = 6, y = 5 and x + y = 11.
If x = 6 and x – y = -7, then we have x = 6, y = 13 and x + y = 19.

Since both conditions together do not 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.

[GMAT math practice question]

(Algebra) xyz≠0. What is the value of x^2 + y^2 + z^2?

1) x + y + z = 3.
2) x^2(1/y+1/z)+y^2(1/z+1/x)+z^2(1/x+1/y)=-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.
Visit https://www.mathrevolution.com/gmat/lesson for details.

Since we have 3 variables (x, y, and z) 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)

x^2(1/y+1/z)+y^2(1/z+1/x)+z^2(1/x+1/y) = -3
=> x^2(1/y+1/z)+y^2(1/z+1/x)+z^2(1/x+1/y) + 3 = 0
=> x^2(1/y+1/z)+y^2(1/z+1/x)+z^2(1/x+1/y) + (x + y + z) = 0
=> x^2(1/x+1/y+1/z)+y^2(1/x+1/y+1/z)+z^2(1/x+1/y+1/z) = 0
=> (x^2+y^2+z^2)(1/x+1/y+1/z) = 0
Then we have 1/x+1/y+1/z=0, since x^2+y^2+z^2≠0.
We have 1/x+1/y+1/z=xy+yz+zx/xyz=0 or xy + yz + zx = 0.
Then x^2 + y^2 + z^2 = (x + y + z)^2 – 2(xy + yz + zx) = 3^2 – 0 = 9.

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

[GMAT math practice question]

(Geometry) The figure shows a right triangle ABC with AB = 8, BC = 10, and three semi-circles with diameters AB, AC, and BC. What is the area of the shaded region?

A. 22
B. 23
C. 24
D. 25
E. 26



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Since triangle ABC is a right triangle, we have AC = 6 with AC^2 = BC^2 - AB^2.
The area of the semi-circle with diameter AB is 4^2π/2 = 8π and the area of the semi-circle with diameter AC is 3^2π/2 = 9π/2 and the area of the semi-circle with diameter BC is 5^2π/2 = 25π/2. The area of triangle ABC is (1/2)6*8 = 24.

We can get the area of the shaded region by subtracting the area of the semi-circle with diameter BC from the sum of the area of triangle ABC and the areas of the semi-circles with diameters AB and AC.

Then the area of the shaded region is 24 + 8π +(9/2) π – (25/2) π = 24.

Therefore, C is the answer.
Answer: C

[GMAT math practice question]

(Algebra) What is the value of 2x + y?

1) (x - y)(y + 3)/4(x - y)^2 +(y + 3)^2= - 1/4
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.
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 (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.

Assume a = x – y and b = y + 3.
When we try to check both conditions together, we realize condition 1) alone is sufficient because condition 1) tells us 2x – y = -3 for the following reason.
(x-y)(y+3)/4(x-y)^2+(y+3)^2= - 1/4
⇔ ab/4a^2+b^2=-14 (a = (x – y) and b = (y + 3))
⇔ 4a^2+b^2=-4ab (cross multiplying)
⇔ 4a^2+4ab+b^2=0 (adding 4ab to both sides)
⇔(2a+b)^2=0 (factoring)
⇔2a+b=0
⇔ 2(x - y) + (y + 3) = 0 (substituting (x – y) and (y + 3) back in)
⇔ 2x – 2y + y + 3 = 0 (multiplying 2 through the first bracket)
⇔ 2x – y + 3 = 0 (adding like terms)
Thus, we have 2x – y = -3 from condition 1) alone.

Condition 2)
Condition 2) is not sufficient, obviously, since condition 2) does not yield a unique solution.

Therefore, A is the answer.
Answer: A

This question is an application of CMT 4(B): condition 2) is easy to work with, and condition 2) is difficult to work with. For CMT 4(B) questions, we may assume condition 1) is sufficient. Since we can figure out condition 2) is not sufficient, we should be able to choose 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.

[GMAT math practice question]

(Geometry) The figure shows a quadrangle ABCD with AB = 3, AD = 5 and CD = 6. The diagonals AC and BD are perpendicular to each other. What is the length of BC?



A. √3
B. 4
C. 2√5
D. 3√3
E. 3√5



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Assume BC = x and P is the intersection of AC and BD.
Then AP^2 + DP^2 = 25, BP^2 + CP^2 = x^2 and we add those two equations and have AP^2 + BP^2 + CP^2 + DP^2 = x^2 + 25.
And AP^2 + BP^2 = 9 and CP^2 + DP^2 = 36 and we add those two equations and have AP^2 + BP^2 + CP^2 + DP^2 = 45.
Thus, we have x^2 + 25 = 45 or x^2 = 20.
Then we have x = 2 √5, and the answer is C.
Answer: C



Therefore, C is the answer.

[GMAT math practice question]

(Number Properties) a, b and c are positive integers. Is 2(a^4+b^4+c^4) a perfect square?

1) a + b + c = 0
2) abc = 6

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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.

Condition 2)
Condition 2) tells us that abc = 6. For three numbers to multiply to 6, the numbers must be 1, 2, and 3 and the order does not matter. We can substitute these numbers into the given equation to get:
2(a^4 + b^4 + c^4) = 2(1^4 + 2^4 + 3^4) = 2(1 + 16 + 81) = 2*98 = 196 = 14^2, which is a perfect square.
Thus condition 2) is sufficient.

Condition 1)
(a + b + c)^2 = a^2 + b^2 + c^2 + 2(ab + bc + ca)
Then, we have a^2 + b^2 + c^2 = -2(ab + bc + ca) since a + b + c = 0.
We have (a^2 + b^2 + c^2)^2 = (-2(ab + bc + ca))2 when we square both sides.
Its left-hand side is
(a^2 + b^2 + c^2)^2
= (a^2 + b^2 + c^2) (a^2 + b^2 + c^2)
= a^4 + a^2b^2 +a^2c^2 + a^2b^2 + b^4 + b^2c^2 + a^2c^2 + b^2c^2 + c^4
= a^4 + b^4 + c^4 + 2a^2b^2 + 2b^2c^2 + 2c^2a^2.

Its right-hand side is
(-2(ab + bc + -ca))^2
= 4(ab + bc + ca)(ab + bc + ca)
= 4(a^2b^2+ ab^2c + a^2bc + ab^2c + b^2c^2 + abc^2 + a^2bc + abc^2 + c^2a^2
= 4(a^2b^2 + b^2c^2 + c^2a^2 + 2a^2bc + 2ab^2c + 2abc^2)
= 4(a^2b^2 + b^2c^2 + c^2a^2) + 2abc(a + b + c))
= 4(a^2b^2 + b^2c^2 + c^2a^2), since a + b + c = 0
We have a^4 + b^4 + c^4 = 2(a^2b^2 + b^2c^2 + c^2a^2).
Then 2(a^4 + b^4 + c^4) = 4(a^2b^2 + b^2c^2 + c^2a^2) = (-2(ab + bc + ca))^2
Thus 2(a^4 + b^4 + c^4) is a square of -2(ab + bc + ca).
We realize we have only used condition 1), but haven’t used condition 2).
It means condition 1) alone is sufficient.
Thus, condition 1) is sufficient.

Therefore, D is the answer.
Answer: D

This question is a CMT 4(B) question: condition 2) is easy to work with, and condition 1) is difficult to work with. For CMT 4(B) questions, D is most likely the answer.

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.