The Ultimate Q51 Guide [Expert Level]

\(4^-^4/16^-^3=?\)

A. 4 B. 1/4 C. 8 D. 1/8 E. 16

==> \(4^-^4=(2^2)^-^4=2^-^8\) and \(16^-^3=(2^4)^-^3=2^-^1^2\) is derived.
Then, \(4^-^4/16^-^3=2^-^8/2^-^1^2=2^-^8^-^(^-^1^2^)=2^4=16\).

Hence, the answer is E.
Answer: E

When a positive integer n has 4 different factors, n=?
1) n has only 1 prime factor
2) n<10

==> In the original condition, there is 1 variable(n), which should match with the number of equations. Then, 1 euqation is needed as well. For 1) 1 equation, for 2) 1 equation, which is likely to make D the answer.
For 1), \(n=2^3, 3^3\),…, which is not sufficient.
For 2), only n=\(2^3\) is possible, which is unique and sufficient.
Hence, the answer is B.
Answer: B

If rs>0, which of the following cannot be true?
A. r>0 B. s>0 C. r+s>0 D. r+s<0 E. r<0<s

==> With 0 in the middle, r and s has to be at the same place, E. r<0<s is impossible. Therefore, the answer is E.
Answer: E

Is \(x^y>1\)?
1) x>1
2) y<1

==> 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)=(2,1/2) yes, but (x,y)=(2,-2) no, NOT sufficient.

Therefore, the answer is E.
Answer: E

Is |x2-y2|<15?
1) |x-y|<3
2) |x+y|<5

==> If you modify the original condition and the question, you get \(-15<x^2-y^2<15?\), -15<(x-y)(x+y)<15? 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 -3<x-y<3 and -5<x+y<5 to -15<(x-y)(x+y)<15, it is always yes, hence it is sufficient. Therefore, the answer is C.
Answer: C

What number is 23 more than three-fourth of itself?

A. 82
B. 92
C. 108
D. 116
E. 124

==> From n=(3/4)n+23, you get 4n=3n+92, hence n=92.

Therefore, the answer is B.
Answer: B

If the average (arithmetic mean) of 5 consecutive multiples of 5 is 30, what is the smallest number?

A. 5
B. 10
C. 15
D. 20
E. 25

==> From 5n,5n+5, 5n+10, 5n+15, 5n+20, the average is 5n+10, and from 5n+10=30, you get 5n=20, n=4. The smallest number=5n=5(4)=20, hence the answer is D.

Answer: D

If \(n=2^23^35^37\), how many factors of n are there?

A. 18
B. 36
C. 54
D. 96
E. 108

==> The number of factors becomes (2+1)(3+1)(3+1)(1+1)=96, hence the answer is D.
Answer: D

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

Is x>y?

1) x+a>x-a
2) ax>ay

==> 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

If m and n are integers greater than 1, mn=?
1) \(m^n=16\)
2) \(m=2\)

==>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 mn=16=24=42, 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

If 1 male, 2 females, and 1 child are to be randomly selected from 8 males, 10 females, and 8 children, how many such cases are possible?
A. 980 B. 1,440 C. 1,880 D. 2,480 E. 2,880

==> You get 8C1*10C2*8C1=(8)(45)(8)=2,880.

The answer is E.
Answer: E

If x and y are positive integers, is xm+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

What is the difference between the circumferences of 2 circles such that the diameters of the circles are 10 and 11?

A. \(\frac{π}{2}\)
B. \(π\)
C. \(\frac{3π}{2}\)
D. \(2π\)
E. \(\frac{5π}{2}\)

==> The circumference of a circle is πd, thus \(π(11-10)= π\). Therefore, the answer is B.
Answer: B

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

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.

Therefore, the answer is B.

What is the difference between the hypotenuse’s length of the right triangle with 2 shorter sides of 10 and 24 and the hypotenuse’s length of the right triangle with 2 shorter sides of 7 and 24?

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

==>For Pythagorean Theorem, 5:12:13=10:24:26 and 7:24:25 appear most frequently. Thus, the length of hypotenuse each becomes 26 and 25, and the difference becomes 26-25=1.

The answer is A.
Answer: A