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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

√5 x √7

Answer 5

√ 35

Question 6

√ 33 = √ a * √ b

Answer 6

√ 11 * √ 3

Question 7

√0.0004

Answer 7

√ 2^2 / √ 10^4 = 2 / 100 = 0.02

Question 8

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

x = ?

Answer 8

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

Question 9

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

Answer 9

x = 0.98 y

Question 10

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

Answer 10

x = 0.4 y

Question 11

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

Answer 11

7 + 12 / 2 = 19 / 2 = 9.5

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

Question 12

1, 2 ,3 . . . 101

Median = ?

Answer 12

101+1 / 2 = 51

When n is odd # (101)

Question 13

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

Answer 13

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

Question 14

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 14

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 15

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

Answer 15

mT:mS - pT:pS

Question 16

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 16

P(GGBB)= 4! / 2! 2! = 6
or 4C2 = 6

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

Therefore, 6/16 = 3/8

Question 17

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

Answer 17

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 18

5 < A < 10 and 2R < A. R의 범위는?

Answer 18

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

Question 19

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

Answer 19

Depends… (not sufficient)

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

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

Question 20

(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 20

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%
(1 million - 10) * 12% > 1 millon * 10%

Question 21

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 21

|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 22

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

Answer 22

y=0
x^4 = 100
x^2 = 10
√x = 3.xx

between 3 and 4

Question 23

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 23

8C2 = 8 * 7 / 2 = 28

Question 24

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 24

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

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

**Why 2^2 instead of 2? **

The Key Idea: LCM includes highest powers of prime factors needed.

If you used only 2 × 3 × 5 × 7 = 210, 210÷4 not divisible

Question 25

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 25

30/3 = 10 30/3^2 = 3 30/3^3 = 1 10 + 3 + 1 = 14

Question 26

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 26

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 27

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

Answer 27

x^2 -2x -1 = 0 x = a and b -> x^2 - (a+b) x + ab = 0

Question 28

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 28

1 + 3C1 + 3C2 + 1 = 8

Question 29

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 29

5!/2! - 4! = 36

Question 30

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 30

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

Question 31

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 31

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 32

Explain steps on using a calculator to solve the following: 2570 * 12 - 1125 * 4

Answer 32

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 33

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 33

교집합

Question 34

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 34

1) 홀 + 홀 + 홀 -> (1/2)^3 2) 홀 + 짝 + 짝 -> (1/2)^3 * 3 1/8 + 3/8 = 4/8 = 1/2

Question 35

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 35

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 36

15 is what percent greater than 3?

Answer 36

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

Question 37

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 37

Set the distance = 120 miles

Question 38

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 38

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

Question 39

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 39

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

Question 40

2^10 = ?

Answer 40

1024

Question 41

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

Answer 41

$100 ( 1 + r/100 )^2

Question 42

3/18 = ?

Answer 42

1/6

Question 43

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 43

**(C) 6** y = 3x + 2 = not a multiple of 3 and therefore not a multiple of 6

Question 44

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 44

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

Question 45

If it is true that x > -2 and x < 7, which of the following must be true? A. x > 2 B. x > -7 C. x< 2 D. -7 < x < 2 E. none of the above.

Answer 45

**B. x > -7**

Question 46

Is the median of set S even? (1) Set S is composed of consecutive odd integers (2) The mean of set S is even.

Answer 46

Is the median of set S even? (1) Set S is composed of consecutive odd integers {1, 3, 5, 7} --> median = 3+5/2 = 4 = even. or {1, 3, 5} --> median = 3 = odd. (2) The mean of set S is even. {1, 1, 4} **(1) + (2) : from (1) median = mean and from (2) mean = even. Therefore, median = mean = even**

Question 47

How much greater is the average (arithmetic mean) of the integers in S than the average of the integers in T? T = 2, 4, 6, 8, 10 S = 9, 11, 13, 15, 17, 19, 21, 23, 25, 27

Answer 47

**For any evenly spaced set, median = mean = the average of the first and the last terms** T = 2, 4, **6**, 8, 10 S = 9, 11, 13, 15, 17, 19, 21, 23, 25, 27 9+27/2 = 36/2 = **18**

Question 48

For positive integers a and b, the remainder when a is divided by b is equal to the remainder when b is divided by a. Which of the following could be a value of ab ? I. 24 II. 30 III. 36

Answer 48

**III only** The only way the remainder when a is divided by b is equal to the remainder when b is divided by a is **a = b since then the remainder in both cases is 0.**

Question 49

There are 10 solid-colored balls in a box, including 1 green ball and 1 yellow ball. If 3 of the balls in the box are to be chosen at random, without replacement, what is the probability that the 3 balls chosen will include the green ball but not the yellow ball?

Answer 49

**without replacement** 1. 8C2 * 1C1 / 10C3 2. 8x7x3 /10x9x8

Question 50

For any numbers a and b if a + b - ab =0, what CANNOT be a value of b?

Answer 50

a - ab = - b a ( 1 - b) = - b a = -b / (1-b) therefore, "1" can not be a value of b

Question 51

original value: x what's the value when x is increased by 50% / 100% / 200% / 250% ?

Answer 51

increased by 50% : 1.5x increased by 100% : x + x = 2x increased by 200% : x + 2x = 3x increased by 250% : x + 2.5x = 3.5x

Question 52

original value: x What's the valuse when x is decreased by 50% / 100% / 200% / 250% ?

Answer 52

decreased by 50% : 0.5x decreased by 100% : x - x = 0 decreased by 200% : x - 2x = - x decreased by 250% : x - 2.5x = -1.5x

Question 53

2 x 4 x 6 …. X 100 = (2x1) x (2x2) …. x (2x50) = ?

Answer 53

= 2^50 x 50! (O) = 2 x 50! (X)

Question 54

Any two consecutive numbers will only share the factor of ____

Answer 54

"1" ex) 99, 100 11, 12

Question 55

T = { 1/2, -1/4, 1/8, -1/16 … } What is the sum of T?

Answer 55

The sum of T = 1/2 ÷ 3/2 = 1/3 first term: 1/2 common ratio: - 1/2 The sum of "infinite geometric progression" with common ratio |r|<1, b ÷ 1 - r, where b is first term.

Question 56

Two thirds of the roads from A to B are at least 5 miles long, and 1/4 of the roads from B to C are at least 5 miles long. If you randomly pick a road from A to B and then randomly pick a road from B to C, what is the probability that at least one of the roads you pick is at least 5 miles long?

Answer 56

1) 4 + 4 + 1 / 12 = 9 / 12 2) 1 - none of the roads you pick will be at least 5 miles long = 1 - 1/3 * 3/4 = 1 - 3 / 12 = 9 / 12

Question 57

What is the largest value of non negative integer N for which 10^N is a factor of 50!?

Answer 57

50/(5^1)+50/(5^2) =10+2 =12

Question 58

Tanya prepared 4 different letters to be sent to 4 different addresses. For each letter, she prepared an envelope with its correct address. If the 4 letters are to be put into the 4 envelopes at random, what is the probability that only 1 letter will be put into the envelope with its correct address?

Answer 58

Probability that first letter in the right envelope= 1/4 Probability that second letter in wrong envelope = 2/3 Probability that third letter in wrong envelope = 1/2 Probability that forth letter in wrong envelope = 1 1/4 * 2/3 * 1/2 = 1/12 1/12 * 4 = 1/3

Question 59

is 0 divisible by 5?

Answer 59

**YES, because 0 = 5 * K, when K = 0** **0 is divisible by EVERY integer.**

Question 60

미지수 2개, 식 1개 두 미지수 값을 구할 수 있음?

Answer 60

**It Depends** 미지수가 2개여도 식 하나로 미지수 값을 구할 수 있음. Ex) 23x + 21y = 130 x = 2, y = 6

Question 61

미지수 2개, 식 2개 두 미지수 값을 구할 수 있음?

Answer 61

**It Depends** 미지수 2개, 식 2개여도 각 미지수의 값을 못 구하는 경우가 있음. ex) 식 2개가 결국 같은 정보를 줄 때. 가능하면 확인 사살

Question 62

Six students are equally divided into 3 groups, then, the three groups were assigned to three different topics. How many different arrangements are possible?

Answer 62

**Six students are equally divided into 3 groups: (6c2 * 4c2 * 2c2)/3!** reason we are dividing by 3! is, to eliminate this {AB, CD, EF} {AB, EF,CD}{CD,AB,EF}{CD,EF,AB}{EF,AB,CD}{EF,CD,AB} **The three groups were assigned to three different topics: 3!** ** = 6c2 * 4c2 * 2c2 = 90**

Question 63

I x I > a, 절대값을 풀면?

Answer 63

x > a or x < -a

Question 64

The probability that a visitor at the mall buys a pack of candy is 30%. If three visitors come to the mall today, what is the probability that exactly two visitors will buy a pack of candy?

Answer 64

3C2 * 0.3 * 0.3 * 0.7 Because the question is not asking about a specific sequence like: **"The first two buy candy, and the third doesn't"** But asking, **"Exactly 2 out of the 3 buy candy — regardless of which ones"** That means there are multiple sequences that satisfy the condition: Buy, Buy, No Buy Buy, No Buy, Buy No Buy, Buy, Buy

Question 65

6 hours 40 mins 을 분수로 표현하시오

Answer 65

= 6 * 2/3 (X) **= 6 + 2/3 (O)**

Question 66

Jin is starting a new job in HR. She might join two projects with possibilities as following: Project A : 1/2 Project B : 3/4 She doesn't know if being in one group affects her chance of being in the other. Question: What's the greatest and least possible chance she joins both groups?

Answer 66

greatest: 1/2 least: 3/4 - 1/2 = 1/4

Question 67

In a certain bathtub, both the cold-water and the hot-water fixtures leak. The cold-water leak alone would fill an empty bucket in c hours and the hot-water leak alone would fill the same bucket in h hours, where c < h. If both fixtures began to leak at the same time into the empty bucket at their respective constant rates and consequently it took t hours to fill the bucket, which of the following must be true? - c/2 < t < h/2

Answer 67

Must be ture ex) c =8, h =10 ( c < h ) is c/2 < t < h/2 ? is 4 < t < 5 ? if c = h = 10 --- t = 5 if c = h = 8 --- t = 4 if c =8, h =10 --- t = 4.x therefore, 4 < t < 5

Question 68

A company produces a certain toy in only 2 sizes, small or large, and in only 2 colors, red or green. If, for each size, there are equal numbers of red and green toys in a certain production lot, what fraction of the total number of green toys is large?

Answer 68

Small - Red: a / Green: a Large - Red: b / Green: b

Question 69

Leo can buy a certain computer for p1 dollars in State A, where the sales tax is t1 percent, or he can buy the same computer for p2 dollars in State B, where the sales tax is t2 percent. Is the total cost of the computer greater in State A than in State B? A: p1 + p1 * t1 B: p2 + p2 * t2 (1) t1 > t2 (2) p1 * t1 > p2 * t2

Answer 69

**(E)** (1)+ (2) - still not sufficient because it could be t1 > 0 and t2 = 0

Question 70

Data sufficiency Q: is A greater than 1? When it's so easy to show/prove that A can be greater than 1, you should try to show/prove that ____________________________.

Answer 70

When it's so easy to show/prove that A can be greater than 1, you should try to show/prove that it is possible that A can be the same or less than 1 . If you can show/prove that, then it is insufficient. easy (이것) - mark hard (저것) - mark

Question 71

There are 27 different three-digit integers that can be formed using only the digits 1, 2 and 3. If all 27 of the integers were listed, what would their sum be?

Answer 71

(100 + 200 + 300) x 9 + (10 + 20 + 30) x 9 + (1 + 2 + 3) x 9

Question 72

Which of the following CANNOT be the least common multiple of two positive integers x and y? (A) xy (B) x (C) y (D) x - y (E) x + y

Answer 72

**(D) x - y** The LCM cannot be smaller than either x or y. x-y is smaller than x

Question 73

If a town of 25,000 people is growing at a rate of approximately 1 percent per year, the population of the town in 5 years will be closest to A. 26000 B. 27000 C. 28000 D. 29000 E. 30000

Answer 73

**1% is very small, so even simple interest calculations will give results close to compound interest.** 1) 25000 * 1.01 ^ 5 = 26,275 2) 25000×0.01×5 = 1250 -> 26,250

Question 74

Don drove x miles. He drove 5 miles at 30 mph and the rest at 60 mph. His travel time was what percent greater than if he had driven the whole trip at 60 mph?

Answer 74

time = distance / speed 1) 5/30 + x-5/60 = x+5/60 2) x/60 **what percent greater (percentage change)** x+5/60 = x/60 * K (X) 5/60 ÷ x/60 = 5/x --- 500/x % (O) ex) 14 is what percent greater than 7? 14-7 (difference) / 7 (original amount) = 1 --- 100% (O) 14=7 * 2 --- 200* (X)

Question 75

Three interviewers, A, B, and C are interviewing 40 applicants. Only with three interviewers' admission can an applicant be admitted. If interviewer A admitted 15 applicants, B admitted 17 applicants, and C admitted 20 applicants, at least how many applicants get the admission?

Answer 75

**(A) 0** 15가 17 안으로 들어가는 경우 세 교집합의 최소는 0 (17+20 < 40)

Question 76

In a village of 100 households, 75 have at least one DVD player, 80 have at least one cell phone, and 55 have at least one MP3 player. If x and y are respectively the greatest and lowest possible number of households that have all three of these devices, x – y is:

Answer 76

최대: 55 최소: 10 1) can solve using bars 2) 100 - 75 = 25 households do not have a DVD player. 100 - 80 = 20 households do not have a cell phone. 100 - 55 = 45 households do not have an MP3 player. 25+20+45=90 90 households might be missing at least one of the three devices. Therefore, the minimum number of households that could have all three devices is: 100−90=10

Question 77

2001, the closing price of stock A and stock B are the same. 2002, the closing price of two stocks both increased and price of A increased more than price of B. By what percent was the price increase of stock A greater than the price increase of stock B?

Answer 77

= a - b / b (O) = a - b / x + b (X)

Question 78

During a 40-mile trip, Marla traveled at an average speed of x miles per hour for the first y miles of the trip and and at an average speed of 1.25x miles per hour for the last 40 - y miles of the trip. The time that Marla took to travel the 40 miles was what percent of the time it would have taken her if she had traveled at an average speed of x miles per hour for the entire trip? (1) x = 48. (2) y = 20.

Answer 78

y / x + 40-y / 1.2x = 40 / x * K = y + 40-y / 1.2 - 40 *K -> y값을 알면 K 값을 알 수 있음

Question 79

If the drama club and music club is combined, what percent of the combined membership is male? (1) Of the 16 members of the drama club, 15 are male. (2) Of the 20 members of the music club, 10 are male.

Answer 79

No C is not right. We cannot assume that there is no overlap between the clubs.

Question 80

a is what fraction of b?

Answer 80

a/b

Question 81

­Consider all positive integers a, b, and c such that 135 is a factor of each of the expressions 15a, 25b and 30c. What is the greatest common factor of all possible sums a + b + c ?

Answer 81

135 = 3^3 * 5 15a = 135 * sth -> a = 9의 배수 = 9x 25b = 135 * sth -> b = 27의 배수 = 27y 30c = 135 * sth -> c = 9의 배수 = 9z 9 (x + 3y + z) ---> 9 is the greatest common factor of a +b + c

Question 82

if |x| > 3, which of the following must be true? I. x > 3 II. x^2 > 9 III. |x - 1| > 2

Answer 82

II and III If A, then B --> A in-circle, B out-circle

Question 83

In a class of 20 students, the average weight of the students is 40 kg and the range is 10. However, the average weight increases to 41 when a new student joins the class in the new session. Based on the above information, select for Maximum range the maximum possible range of the weight of the new class and for Minimum range the minimum possible range of the weight of the new class.

Answer 83

Min: 21.5 Max: 30.5 https://gmatclub.com/forum/i12-437845.html