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Question 1

A number is divisible by 3 if __________________

Answer 1

A number is divisible by 3 if the sum of all the digits is divisible by 3

ex) 531

Question 2

A number is divisible by 6 if __________________

Answer 2

A number is divisible by 6 if the number is divisible by both 2 and 3

cf. a number is divisible by 2 if the ones digit is 0, 2, 4, 6 or 8

ex) 174

Question 3

A number is divisible by 9 if __________________

Answer 3

A number is divisible by 9 if the sum of all the digits is divisible by 9

ex) 333

Question 4

The number of factors of 240

Answer 4

240 = 2^4 x 3^1 x 5^1
-> (4+1) x (1+1) x (1+1) = 20
240 has a total of 20 factors

1, 2, 3, 4 …….. 60, 80, 120, 240

Question 5

x, y 의 최대공약수 x 최소공배수 = ?

The greatest common factor (GCF)
The lowest common multiple (LCM)

Answer 5

xy

Question 6

√5 x √7

Answer 6

√ 35

Question 7

√ 33 = √ a * √ b

Answer 7

√ 11 * √ 3

Question 8

√0.0004

Answer 8

√ 2^2 / √ 10^4 = 2 / 100 = 0.02

Question 9

|16x+14|=|8x+6|

x = ?

Answer 9

1) 16x + 14 = 8x +6 —–> x = -1
2) 16x + 14 = -(8x+6) —–> x = -5/6

Question 10

x is 2% less than y
x = k y
k=?

Answer 10

x = 0.98 y

Question 11

x is 60% less than y
x = k y
k=?

Answer 11

x = 0.4 y

Question 12

the average of 7, 8, 9, 10, 11, 12

Answer 12

7 + 12 / 2 = 19 / 2 = 9.5

average in a set of consecutive integers =
highest # + lowest # / 2

Question 13

1, 2 ,3 . . . 101

Median = ?

Answer 13

101+1 / 2 = 51

When n is odd # (101)

Question 14

1, 2 ,3 . . . 100
Median = ?

Answer 14

100 / 2 = 50 and 51
50+51 / 2 = 50.5

Question 15

A straight pipe 1 yard in length was marked off in fourths and also in thirds. If the pipe was then cut into separate pieces at each of these markings, which of the following gives all the different lengths of the pieces, in fractions of a yard?

Answer 15

Take the length of the pipe to be 12-meter long (the least common multiple of 3 and 4.)

In this case the branch would be cut at 3, 4, 6, 8, and 9 meters.

3/12=1/4
1/12
2/12=1/6

Question 16

The ratio of the number of teachers to the number of students is the same in School District M and School District P.

Answer 16

mT:mS - pT:pS

Question 17

A couple decides to have 4 children. If they succeed in having 4 children and each child is equally likely to be a boy or a girl, what is the probability that they will have exactly 2 girls and 2 boys?

Answer 17

4C2/2^4 = 6/16 = 3/8

ㅁㅁㅁㅁ
2 x 2 x 2 x2 = 16 total (boy or girl)
4C2 = 6

Question 18

If 0 < r < 1 < s < 2.
rs must be less than 1?

Answer 18

No.
ex) r=0.99, s=1.99

숫자를 대입해서 풀 때, 범위 안에서 최대한 극단적인 숫자를 선택.
Ex) 0<x<1, 1/2를 넣어볼 수도 있지만, 0.99 or 0.01 을 넣어서 확인 사살.

Question 19

5 < A < 10 and 2R < A. R의 범위는?

Answer 19

2R < A < 10
=> 2R < 10
=> R < 5

Question 20

a+30/b+30 is greater than a/b?

Answer 20

Depends… (not sufficient)

If a=1 and b=2, 31/32 > 1/2

But if a=2 and b= 1, 32/31 < 2

Question 21

(1) apple price is $10 less than orange price
(2) apple discount % is 2% greater than the orange discount %

which one is discounted by the greater dollar amount?

Answer 21

We don’t know

if apple is $10 and orange is $20
vs if apple is $990 and orange is 1000

10 * 12% < 20 * 10%
990 * 12% > 1000 * 10%

Question 22

In planning for a car trip, Joan estimated both the distance of the trip, in miles, and her average speed, in miles per hour. She accurately divided her estimated distance by her estimated average speed to obtain an estimate for the time, in hours, that the trip would take. Was her estimate within 0.5 hours of the actual time that the trip took?

(1) Joan’s estimate for the distance was within 5 miles of the actual distance.

(2) Joan’s estimate for her average speed was within 10 miles per hour of her actual average speed.

The only thing we need to prove is ?

Answer 22

|ET-AT| < 0.5 hours?

if ET = AT, |ET-AT| = 0.

Therefore, only thing we need to prove is |ET-AT| > 0.5 hours.

T = D / S

ET = 20 / 30 = 1.5
AT = 15 / 40 = 0.375 (to maximize, AT should be minimized as possible)
ET - AT > 1

Question 23

If x^4 + y^4 = 100, then the greatest possible value of x is between

Answer 23

y=0
x^4 = 100
x^2 = 10
√x = 3.xx

between 3 and 4

Question 24

There are 8 teams in a certain league and each team plays each of the other teams exactly once. If each game is played by 2 teams, what is the total number of games played?

Answer 24

8C2 = 8 * 7 / 2 = 28

Question 25

What is the lowest positive integer that is divisible by each of the integers 1 through 7, inclusive?

(A) 420
(B) 840
(C) 1,260
(D) 2,520
(E) 5,040

Answer 25

The integer should be divisible
by: 2, 3, 4(=2^2), 5, 6(=2 * 3), and 7.

LCM= 2^2 * 3 * 5 * 7 = 420

Question 26

If p is the product of the integers from 1 to 30, inclusive, what is the greatest integer k for which 3^k is a factor of p?

Answer 26

30/3 = 10
30/3^2 = 3
30/3^3 = 1

10 + 3 + 1 = 14

Question 27

A store currently charges the same price for each towel that it sells. If the current price of each towel were to be increased by $1, 10 fewer of the towels could be bought for $120, excluding sales tax. What is the current price of each towel?

(A) $ 1
(B) $ 2
(C) $ 3
(D) $ 4
(E) $12

Answer 27

P * N = 120
(P+1) * (N-10) = 120

“When substituting answer choices, it’s good to start with the middle value”

(C) $ 3 - P

Question 28

what equation has 1 + √2 as one of its roots?

Answer 28

x^2 -2x -1 = 0

x = a and b
-> x^2 - (a+b) x + ab = 0

Question 29

The subsets of the set {w, x, y} are {w}, {x}, {y}, {w, x}, {w, y}, {x, y}, {w, x,y}, and { } (the empty subset). How many subsets of the set {w, x, y, z} contain w ?

Answer 29

1 + 3C1 + 3C2 + 1 = 8

Question 30

The letters D, G, I, I , and T can be used to form 5-letter strings as DIGIT or DGIIT. Using these letters, how many 5-letter strings can be formed in which the two occurrences of the letter I are separated by at least one other letter?

Answer 30

5!/2! - 4! = 36

Question 31

If n = 3^8 - 2^8, which of the following is NOT a factor of n?

(A) 97
(B) 65
(C) 35
(D) 13
(E) 5

Answer 31

apply a^2 - b^2 = (a-b)(a+b)

Question 32

If n is a positive integer and n^2 is divisible by 72, then the largest positive integer that must divide n is

A. 6
B. 12
C. 24
D. 36
E. 48

Answer 32

n^2 = 72 * 몫
= 6^2 * 2 * 몫

n = 6 * √ 2 * 몫
n의 최소값 = 12

n의 값은 무수히 커질 수 있다. n의 최소값은 12. Must는 최소값을 나타낸다.

Question 33

Explain steps on using a calculator to solve the following:

2570 * 12 - 1125 * 4

Answer 33

2570 * 12 = value M+
1125 * 4 = value M-
And then press MRC once.
(The display will show 26,340, the final result.)
Press MRC twice to clear any previous memory.

Question 34

홀 + 홀 =
홀 + 짝 =
짝 + 짝 =

홀 x 홀 =
홀 x 짝 =
짝 x 짝 =

Answer 34

홀 + 홀 = 짝
홀 + 짝 = 홀
짝 + 짝 = 짝

홀 x 홀 = 홀
홀 x 짝 = 짝
짝 x 짝 = 짝

Question 35

Did most respondents who indicated they somewhat agreed that class sizes should be reduced also indicate that they somewhat agreed that music classes should be added?

What does “also” indicate?

Answer 35

교집합

Question 36

A box contains 100 balls, numbered from 1 to 100. If three balls are selected at random and with replacement from the box, what is the probability that the sum of the three numbers on the balls selected from the box will be odd?

Answer 36

1) 홀 + 홀 + 홀 -> (1/2)^3

2) 홀 + 짝 + 짝 -> (1/2)^3 * 3

1/8 + 3/8 = 4/8 = 1/2

Question 37

A certain club has 10 members, including Harry. One of the 10 members is to be chosen at random to be the president, one of the remaining 9 members is to be chosen at random to be the secretary, and one of the remaining 8 members is to be chosen at random to be the treasurer. What is the probability that Harry will be either the member chosen to be the secretary or the member chosen to be the treasurer?

Answer 37

First, draw the box!
ㅁ ㅁ ㅁ
P S T
Total: 10 * 9 * 8
Harry=S: 9 * 1 * 8
Harry=T: 9 * 8 * 1

72 * 2 / 10 * 9 * 8 =1 / 5

Question 38

15 is what percent greater than 3?

Answer 38

= percentage increase from 3 to 15
= 15 - 3 / 3 * 100%
= 12 / 3 * 100%
= 400%

Question 39

A car travels from Mayville to Rome at an average speed of 30 miles per hour and returns immediately along the same route at an average speed of 40 miles per hour. Of the following, which is closest to the average speed, in miles per hour, for the round trip?

Answer 39

Set the distance = 120 miles

Question 40

If a committee of 3 people is to be selected from among 5 married couples so that the committee does not include two people who are married to each other, how many such committees are possible?

Answer 40

5C3 * 2^3 = 10 * 8 = 80

Question 41

Jack is now 14 years older than Bill. If in 10 years Jack will be twice as old as Bill, how old will Jack be in 5 years?

Answer 41

J = B + 14
J + 10 = (B + 24) x 2 (X)
J +10 = (B + 10) x 2 (O)

Question 42

2^10 = ?

Answer 42

1024

Question 43

is the following correct?
2^12x4 = 2^12 x 12^4

Answer 43

= 2^12 x 12^4 = 2^16 (2^a x 2^b = 2^a+b)
= 2^48 = 2 ^12 * 2^36 (O)

Question 44

$100, r% compounded annually.
What’s 원금 + 이자 in 2 years after investment?

Answer 44

$100 ( 1 + r/100 )^2

Question 45

3/18 = ?

Answer 45

1/6

Question 46

If x is an integer and y = 3x + 2, which of the following CANNOT be a divisor of y?

(A) 4
(B) 5
(C) 6
(D) 7
(E) 8

Answer 46

(C) 6

y = 3x + 2 = not a multiple of 3 and therefore not a multiple of 6

Question 47

Tom, Jane, and Sue each purchased a new house. The average (arithmetic mean) price of the three houses was $120,000. What was the median price of the three houses?

(A) The price of Jane’s house was $120,000.

Is (A) sufficient to know the median price?

Answer 47

Is (A) sufficient to know the median price? YES
Tom + Sue = 24
ex) 10, 12, 14
ex) 12, 12, 12