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==> In the original condition, there are 2 variables (a,b) and in order to match the number of variables to the number of equations, there must be 2 equations. Since there is 1 for con 1) and 1 for con 2), C is most likely to be the answer. By solving con 1) and con 2), from a=±b, (a,b)=(1,1) yes, but (a,b)=(1,-1) no, hence it is not sufficient.

Therefore, the answer is E. Answer: E
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What is the perimeter of a certain right triangle? 1) The length of hypotenuse is 4 2) The area of triangle is 4.5

==> In the original condition, for the right triangle, there are 2 variables and in order to match the number of variables to the number of equations, there must be 2 equations. Since there is 1 for con 1) and 1 for con 2), C is most likely to be the answer. By solving con 1) and con 2) and assume the two hypotenuses of the triangle as a and b, you get \(a^2+b^2=4^2=16\), and from ab/2=4.5, you get ab=9. Then, from \((a+b)^2=a^2+b^2+2ab=16+2(9)=16+18=34\), you get a+b=√34, so the perimeter of the right triangle=a+b+4=√34+4, hence it is unique and sufficient.

Therefore, the answer is C. Answer: C
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When a positive integer n is divided by 2, what is the remainder?

1) The remainder is 1 when n is divided by 5 2) The remainder is 1 when n is divided by 10

==> In the original condition, there is 1 variable (n) and in order to match the number of variables to the number of equations, there must be 1 equation. Since there is 1 for con 1) and 1 for con 2), D is most likely to be the answer. For remainder questions, you can solve it by using direct substitution. For con 1), from n=5p+1(p=any positive integer), you get n=1,6,… However, from n=1=2(0)+1, you get r=1, but n=6=2(3)+0, you get r=0, hence it is not unique and not sufficient. For con 2), from n=10q+1(q=any positive integer), you get n=1,11,21,… However, for all cases, the remainder=1, hence it is unique and sufficient.

==> If you modify the original condition and the question, from \(2^x^+^y/2^x^-^y=2^x^+^y^-^(^x^-^y^)=2^x^+^y^-^x^+^y=2^2^y\), you only need to know y. From con 2) y=2, it is sufficient.

Therefore, the answer is B. Answer: B
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In the x-y plane, If line k does not pass through the origin, is the slope of the line K negative?

1) The y-intercept of the line K is 4 times the x-intercept of the line K 2) The product of the y-intercept and the x-intercept of the line K is positive

==> In the original condition, there are 2 variables(there are 2 variables for a line -> slope and y-intercept). In order to match with the number of equations, you need 2 equations. For 1) 1 equation and for 2) 1 equation, which is likely to make C the answer. Through 1) & 2), 1)=2) is derived and it is yes for each condition.

Hence, it is sufficient and the answer is D. Answer: D
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==> In the original condition, there is 1 variable(a). In order to match with the number of equations, you also need 1 equation. For 1) 1 equation and for 2) 1 equation, which is likely to make D the answer. In case of 1), a=1, 2 is given, which is not unique and not sufficient. In case of 2), only a=-1 is possible, which is unique and sufficient.

Hence, the answer is B. Answer: B
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If m and n are positive integers, what is the value of m+n?

1) m/n=3/5 2) The greatest common divisor of m and n is 5

==>In the original condition, there are 2 variables (m, n) for the right triangle, and in order to match the number of variables to the number of equations, there must be 2 equations. Since there is 1 for con 1) and 1 for con 2), C is most likely to be the answer. By solving con 1) and con 2), you get M=3*5=13 and n=5*5=25, so m+n=25+15=40, hence it is unique and sufficient.

Therefore, the answer is C. Answer: C
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==> In the original condition, there are 2 variables (x, y) and in order to match the number of variables to the number of equations, there must be 2 equations. Since there is 1 for con 1) and 1 for con 2), C is most likely to be the answer. By solving con 1) and con 2), from con 1), you get a>-a, 2a>0, or a>0, and from con 2), you get ax>ay, and the inequality sign doesn’t change even if you divide both sides by a because since a>9, you get x>y, hence yes, it is always sufficient.

Therefore, the answer is C. Answer: C
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==>In the original condition, there are 2 variables (m, n), and in order to match the number of variables to the number of equations, there must be 2 equations. Since there is 1 for con 1) and 1 for con 2), C is most likely to be the answer. By solving con 1) and con 2), you get m=2 and n=4, hence it is sufficient, and the answer is C. However, this is an integer question, one of the key questions, so you apply CMT 4 (A: if you get C too easily, consider A or B). For con 1), from\(m^n=16=2^4=4^2\), you get (m,n)=(2,4),(4,2), which always becomes mn=8, hence it is sufficient.

Therefore, the answer is A, not C. Answer: A
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If x and y are positive integers, is \(x^m+y\) a multiple of 9?

1) m is a multiple of 3 2) x+y is a multiple of 9

==> In the original condition, there are 3 variables (x, y, m) and in order to match the number of variables to the number of equations, there must be 3 equations. Since there is 1 for con 1) and 1 for con 2), E is most likely to be the answer. By solving con 1) and con 2), (x,y,m)=(1,8,3) yes, but (x,y,m)=(2,7,3) no, hence it is not sufficient.

Therefore, the answer is E. Answer: E
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John and Tom leave the school at the same time to each of their houses. Does John reach his house faster than Tom?

1) The distance between the school and John’s house is 10km farther than the distance between the school and Tom’s house 2) Tom’s speed is 80% of John’s speed.

==> In the original condition, there are 5 variables (v1, t1, v2, t2, d) and 2 equations (v1t1=d and v2t2=d), and in order to match the number of variables to the number of equations, there must be 3 more equations. Since there is 1 for con 1) and 1 for con 2), E is most likely to be the answer. By solving con 1) and con 2), you cannot find the value of d, hence it is not sufficient.

==> In the original condition, there is 1 variable (x) and in order to match the number of variables to the number of equations, there must be 1 equation. Since there is 1 for con 1) and 1 for con 2), D is most likely to be the answer. For con 1), it is always yes, hence it is sufficient. For con 2), x=0 yes, but x=-3.5 no, hence it is not sufficient.

Therefore, the answer is A. Answer: A
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If d and n are positive integers, is \(d^n\) an even number?

1) d is divisible by 8 2) n is divisible by 4

==> If you modify the original condition and the question, in order for \(d^n\) to become an even number, d has to be even. For con 1), from d=8t (t=any positive integer), it is always even, hence yes, it is sufficient.

John traveled 150 miles. What is the average speed of John on the trip?

1) John traveled the first 100 miles at the rate of 50 miles per hour 2) John traveled the last 100 miles at the rate of 50 miles per hour

==> In the original condition, he travels the total 150 miles by dividing it to two trips of 50 miles each. Hence, since there are 6 variables, E is most likely to be the answer. In order for C to be the answer, there must be a word “constant rate” mentioned.

Therefore, E is the answer. Answer: E
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For museum X, the total entrance fee is $x plus $z each person older than 30. For museum Y, the total entrance fee is $y plus $w each person older than 40. When 50 people enter museums X and Y, is the total entrance fee of the museum X smaller than that of the museum Y?

1) x<y 2) z<w

==> In the original condition, there are 6 variables (Mx, x, z, My, y, w) and 2 equations (Mx=x+20z and My=y+10w) and in order to match the number of variables to the number of equations, there must be 4 more equations. Since there is 1 for con 1) and 1 for con 2), E is most likely to be the answer. By solving con 1) and con 2), if (x,z,y,w)=(1,1,2,2), from Mx=21 My=22, it is yes, but if (x,z,y,w)=(2,2,3,3), from Mx=42 My=33, it is no, hence it is not sufficient.

Therefore, the answer is E. Answer: E
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==> In the original condition, there are 3 variables (x, y, z) and in order to match the number of variables to the number of equations, there must be 3 equations. Since there is 1 for con 1) and 1 for con 2), E is most likely to be the answer. By solving con 1) and con 2), you get \((a+b+c)^2=a^2+b^2+c^2+2(ab+bc+ca)=56+2(10)=76\), which becomes \(a+b+c=\sqrt{76}\), hence it is unique and sufficient.

Therefore, the answer is C. Answer: C
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==> In the original condition, there are 2 variables (x, y) and in order to match the number of variables to the number of equations, there must be 2 equations. Since there is 1 for con 1) and 1 for con 2), C is most likely to be the answer. By solving con 1) and con 2), you get (x,y)=(1,3),(3,1),(-1,-3),(-3,-1), and all become (x-y)2=4, hence it is unique and sufficient.

Therefore, the answer is C. Answer: C
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==> If you modify the original condition and the question, \((x+y)^2-(x-y)^2=?\) becomes (x+y-x+y)(x+y+x-y)=?, and if you simplify this, you get (2y)(2x)=?, 4xy=?. Thus, for con 1), you get xy=5, hence it is unique and sufficient.