GMAT - Hard Quant - Number Properties

Min number of seats that must be filled to fill the arena

(1) find the max number of seats without filling any (2) total = 150 (3) - 5 = 145 (4) +3 = 148

x = 9^10 - 3^17 and (x/n) is an integer. if n is positive and has exactly 2 factors, how many values of n are possible?

ANS: 3 x = 3^20 - 3^17 CANNOT subtract - must factor 3^17 (3^3-1) = 3^17(26) No need to expand further, find prime factors = 2,3,13

When tossed, a certain coin has equal probability of landing on either side. If the coin is tossed 3 times, what is the probability that it will land on the same side each time? A. 1/8 B. 1/4 C. 1/3 D. 3/8 E. 1/2

1/4 Permutations with replacement 2x2x2 gives you all the possibilities (taking into consideration order) Write out: only 2 distinct sequences fit Or 1/2 x 1/2 x 1/2 = possibility any of the sequences will occur

Which of the following must be an integer if x is a positive integer and 4/x + 5/x + 6/x is an integer?

ANS: 30/x add the terms = 15/x x must be a factor of 15 therefore, 30/ any factor of 15 is an integer

Is x < w ? (1) x - w < x (2) x + w < w

ANS C (1) w>0 (2) x<0

Is the product of x and y greater than the sum of x and y? (1) xy < 0 (2) x > -y

ANS C Test cases (1) If x = 2, y = -3 then x+y >xy If x = .5, y = -2, then x+y<xy (2) if x = 1, y = 1, x+y>xy if x = 3, y = 2, x+y < xy

What is the value of x^2 - y^2 (1) x+y = 2x (2) x-y = 0

ANS D

An investor purchased 100 shares of stock X at 6 1/8 dollars per share and sold them all a year later at 24 dollars per share. If the investor paid a 2 percent brokerage fee on both the total purchase price and the total selling price, which of the following is closest to the investor's percent gain on this investment? (A) 92% (B) 240% (C) 280% (D) 300% (E) 380%

ANS: (c) Cost = $6.125 * (1.02) Revenue = $24*(.98) ratio = 24/6.125 (just under 4x or 300%) Cost is slightly higher, revenue is slightly lower so 280%

A retail item is offered at a discount of p percent (where p > 10), with a 5% state sales tax assessed on the discounted purchase price. If the state sales tax were not assessed, what percent discount from the item's original retail price, in terms of p, would result in the same final price?

ANS: 1.05(p-5) Test smart numbers Algebraic R(100 - p)(1.05) = R(100 - a)

ABC + BCB CDD In the addition shown above, A, B, C, and D represent the nonzero digits of three 3-digit numbers. What is the largest possible value of the product of A and B ?

ANS: 10 step 1 - C+B <=9 because otherwise there will be carry over step 2 - a+b = c so b<c step 3 - to maximize ab, assume a+b = c and b+c = 9 Make grid

If n = 10^10 and n^n = 10^d, what is the value of d?

ANS: 10^11 (10^10)^(10^10) exponent raised to and exponent can be multiplied 10*(10^10) = 10^11

Set S contains seven distinct integers. The median of set S is the integer m, and all values in set S are equal to or less than 2m. What is the highest possible average (arithmetic mean) of all values in set S ?

ANS: 10m/7 - 9/7 The numbers are distinct!

Working alone at it's constant rate, pump X pumped out 1/4 of the water in a tank in 2 hours. Then pumps Y and Z started working and the three pumps, working simultaneously at their respective constant rates, pumped out the rest of the water in 3 hours. If pump Y, working alone at it's constant rate, would have taken 18 hours to pump out the rest of the water, how many hours would it have taken pump Z, working alone at it's constant rate, to pump out all of the water that was pumped out of the tank? A) 6 B) 12 C) 15 D) 18 E) 24

ANS: 12 1. Find rates of x and y Rate = tank / hour Rate x = 1/8 Rate y = (3/4)/18 = 24 2. Solve for z 3(1/8+1/24+1/z) = 3/4 rate x time = tanks

If a, b, and c are different nonnegative digits, which of the following CANNOT be a solution to the addition problem below? abc + cba 929 1,110 1,111 1,322 1,776

ANS: 1322 if a + c >= 10, odd tens digit if a + c < 10, even tens digit Both the ones digit and hundreds digit are formed by summing a + c, so these digits will either be the same (if we don't carry from the tens digit) or differ by exactly 1 (that is, if a 1 is carried from the tens digit).

For any integer k > 1, the term "length of an integer" refers to the number of positive prime factors, not necessarily distinct, whose product is equal to k. For example, if k = 24, the length of k is equal to 4, since 24 = 2 × 2 × 2 × 3. If x and y are positive integers such that x > 1, y > 1, and x + 3y < 1000, what is the maximum possible sum of the length of x and the length of y?

ANS: 16 The greatest number of factors is calculated by using the smallest prime number, 2, as a factor as many times as possible. 29 = 512 and 210 = 1,024, so our largest possible length for x is 9.

Roberto has three children: two girls and a boy. A year from now, the sum of the girl's ages will equal the boy's age. Three years from now, what will be the absolute value of the difference between the age of the boy and the combined ages of the girls?

ANS: 2 We know ages were equal 1 year from now So G = B 3 years from now = 2 years from next year) G+4 = B+2

If x and y are positive integers, which of the following CANNOT be the greatest common divisor of 35x and 20y? 5 5(x - y) 20x 20y 35x

ANS: 20x 35x/20x is NOT an integer

How many of the integers between 1 and 400 inclusive are not divisible by 4 or include 4 as a digit?

ANS: 252 Count multiples of 4 = 100 Reminding = 300 count those with 4 in the hundreds 400, but already removed count those with 4 in the tens - list out (MUST not be a multiple of 4) *last 2 digits multiple of 4 is a multiple of 4 Multiply by 4 (for each "tens") Count those with 4 in the units, multiply by 4

It is known that no more than 7 children will be attending a party. What is the smallest number of cookies that must be brought to the party so that each child receives the same number of cookies and no cookies are left over or broken? 35 105 180 210 420

ANS: 420 LCM question Note how many times each prime factor is needed 2 => 2 times 3 = > 1 time 7 = one time 5 = one time

The average (arithmetic mean) of the multiples of 6 that are greater than 0 and less than 1,000 is 499 500 501 502 503

ANS: 501 Evenly spaced find first and last term and average

Bag A contains red, white and blue marbles such that the red to white marble ratio is 1:3 and the white to blue marble ratio is 2:3. Bag B contains red and white marbles in the ratio of 1:4. Together, the two bags contain 30 white marbles. How many red marbles could be in bag A? A. 1 B. 3 C. 4 D. 6 E. 8

ANS: 6 Given 30 white marbles, so find the constraints of white marbles Wa must be mult of 6 Wb must be mult of 4 Therefore, Wa could be 6 or 18 for Wb to work Ra = 2 or 6

The integers x and y are both positive, the remainder when x is divided by 12 is 7, and the remainder when y is divided by 12 is 3. Each of the following is a possible value of 2x + y EXCEPT 125 101 77 63 53

ANS: 63 First, find what 2x / 12 Remainder is Test: 7 x 2 = 14 Remainder 2 y/12 R3 So 2x + y R 5

the ratio of soap to alcohol to water is 2:50:100 The solution is altered such that the ratio of s:a is doubled while s:w is halved if the altered solution contains 100 alcohol, how much water?

ANS: 800 doubling s:a - increase s to 4 halving s:w Since you increased to 4, w must be 400 to increase ratio

What is the tens digit of positive integer n? (1) Hundreds digit of 10n = 2 (2) tens digit of n - 9 =1

ANS: A (2) does not work because n can be anything from 19 to 28 Method: n - 9 = [10 to 19] n = [19 to 28]

Is the number x positive? (1) On the number line, 0 is closer to x - 1 than to x. (2) On the number line, 0 is closer to x than to x + 1.

ANS: A (2) insufficient because x could still be to the left of 0

If x is a positive integer, is x / 21 an integer? (1) 4x/21 is integer (2) 12x/21 is integer

ANS: A Asking if x is divisible by 21 Use prime factorization (1) 4 and 21 share no prime factors so x must be divisible by 21 (2) 12 and 21 share 3, so x must be divisible by 7 but not necessarily 21

if x and y are positive, is x<y? (1) √x<√y (2) (x−3)^2<(y−3)^2

ANS: A Given root of positive, only take positive!

If Q is a set of consecutive integers, what is the standard deviation of Q? (1) Set Q contains 21 terms. (2) The median of set Q is 20.

ANS: A Sufficient because we know it is a CONSECUTIVE set

This table lists enrollment in an afterschool program by activity. There are 30 total students enrolled in the entire program. Students may participate in one, two, or three activities. How many students participate in all three activities? (1) 21 students participate in only one activity. (2) 6 students participate in both basketball and math.

ANS: A Total in table (not shown) of unique enrollments = 42 > 30 so there are people in more than one (1) 21 students are in 1, so 9 are in 2 or 3 42 enrollments - 21(1) = 21 spots to account for if all 9 were only in 2 each, then they account for 18 of the 21 enrollments. you know that 3 must be in 3 and 6 are in 2

If x and y are non-zero integers and |x| + |y| = 32, what is xy? (1) -4x - 12y = 0 (2) |x| - |y| = 16

ANS: A can get to the answer quickly by solving (2) |x| + |y| = 32 |x| - |y| = 16 |x|= 24 |y|=8 but keep the abs values! Don't know the signs

If N is a positive three-digit number that is greater than 200, and each digit of N is a factor of N itself, what is the value of N? (1) The tens digit of N is 5. (2) The units digit of N is 5.

ANS: A cannot be 155 Test others: 355, 455 etc Must be 555

Each of the following equations has at least one solution EXCEPT -2^n = (-2)^-n 2^-n = (-2)^n 2^n = (-2)^-n (-2)^n = -2^n (-2)^-n = -2^-n

ANS: A when n = 0, will still be opposite signs when n = 1, will be reciprocals

Is a - b > 0? (1) a > 2b (2) b > a + 3

ANS: B

The annual rent collected by a corporation from a certain building was x percent more in 1998 than in 1997 and y percent less in 1999 than in 1998. Was hte annual rent collected from the building more in 1999 than 1997? (1) x > y (2) (xy)/100 < (x-y)

ANS: B (1) Not sufficient because theoretically, the decrease from 1998 - 1999 is on a larger base, so y could be smaller than x for 1999 still to be less than 1997 Algebraically Rent 1999 = (1+ x/100) (1-y/100) A where A = rent 1997, plug numbers (2) sufficient, recognize the quadratic (1+ x/100)(1-y/100)>y

If x is positive, is x prime? (1) x^3 has exactly four distinct positive integer factors. (2) x^2 − x = 6

ANS: B (1) is not sufficient because we don't know that x^3 is a perfect cube if x^3 is 6, which has 4 distinct factors, then cube root x is not even an integer (2) solve for x; x =3

If 3^a4^b = c and each of the variables a, b, and c is an integer, then what is the value of b ? (1) 5^a = 25 (2) c = 36

ANS: B (2) 3^a4^b=36 Though there are two variables, there is only one possibility for b because 36 is not divisible by any other powers of 4!

Stores X, Y, and Z each sell a certain item that has a given list price. Stores X and Y are located in a state with a 5 percent sales tax, and both sell the item at a 5 percent discount off list price, while Store Z is located in a state with no sales tax and gives no discounts. Store X applies its discounts first and then charges sales tax on the discounted price, while Store Y adds the tax first and then applies the discount to the price with tax. If x and y are the prices, with tax and discount, charged by Stores X and Y, respectively, and z is the price charged by Store Z, which of the following statements correctly describes the relationship among x, y, and z? A. x=y=z B. x=y<z C. x<y<z D. x<z<y E. y<z<x

ANS: B 5% tax and 5% discount in whichever order is the same Both are (0.95)(1.05) Don't confuse with comparing to the "no change case", in which taking 5% off of a higher number is going to yield a lower add-back

If K is a positive integer, and n = k(k+7), is n divisible by 6? (1) k is odd (2) when k is divided by 3, remainder = 2

ANS: B N must be divisible by 2 and 3 (1) if k is odd, k+7 must be even, but uncertain if n is divisible by 3 (2) if k / 3 R 2, then k+7 is divisible by 3 If k is even (i.e. 8), then n is divisible by 6 if k is odd (i.e. 5), then (k+7) is even and n is again divisible by 6; this is sufficient

If a, b, and c are positive integers such that 1/a + 1/b = 1/c, what is the value of c? (1) b ≤ 4 (2) ab ≤ 15

ANS: B We cannot assume C can be anything C must be a pos integer so there are only certain combos where 1/a + 1/b = 1/c Test cases (a,b) c (2,2) 1

In a certain sequence, each term, starting with the 3rd term, is found by multiplying the previous two terms. What is the difference between the 6th and 3rd terms in the sequence? (1) The 1st term is equal to 8 times the 2nd term. (2) The 4th term is equal to 1.

ANS: B list out first 6 terms 1 A 2 B 3 AB 4 AB^2 5 A^2B^3 6 A^3B^5 Factor out 4th term AB ((Ab^2)^2 - 1) 1-1=0 OR... 3rd term: n 4th term: 1 5th term: n*1 = n 6th term: 1*n = n Therefore, the difference between them is 0 and the correct answer is B.

If ab ≠ 0, is ab > a/b ? (1) |b| > 1 (2) ab + a/b > 0

ANS: C (1) Translates to b>1 or b <-1 (2) Translates to both are pos or neg Together - test cases both pos or negative means multiplying together is greater than dividing

A certain research group plans to create computer models of x% of a list of 10,000 bacterial species known to inhabit the human body. After a budget cut, the group finds it must reduce this selection by (x − 5)%. In terms of x, how many species of bacteria will the group be able to model? (x+5)/100 x2 - 5x (x)(105 - x) (100)(105 - x) (100)(95 - x)

ANS: C Algebraicly step 1 - (x/100)*10000 = 100x step 2 - (100- (x-5))/100 Multiply 100x(100-(x-5))/100 = (105-x)(x)

If y≠1, is x=1? (1) x^2 + y^2 = 1 (2) y = 1 - x

ANS: C If x = 0, y = 1, and stated that y cannot = 1

Three points T, U and V on the number line have coordinates t, u, and v respectively. Is T between points U and V? (1) t^2 < 4 < u^2 < v^2 (2) u<0<v

ANS: C List out possible values of t t>-2, t<2 U and V are <-1 or >2 Only works if v and u are opposite signs

x is the sum of y consecutive integers. w is the sum of z consecutive integers. If y = 2z, and y and z are both positive integers, then each of the following could be true EXCEPT x = w x > w x/y is an integer w/z is an integer x/z is an integer

ANS: C RULE: For any set of consecutive integers with an odd number of terms, the sum of the integers is always a multiple of the number of terms. For example, the sum of 1, 2, and 3 (three consecutives -- an odd number) is 6, which is a multiple of 3. For any set of consecutive integers with an even number of terms, the sum of the integers is never a multiple of the number of terms. Once you know this, you know y is even so x/y cannot be an integer

From a Group of 8 People, Including George and Nina, 3 people are to be selected at random to work on a certain project. What is the probability that 3 people selected will include George but not Nina A 5/56 B 9/56 C 15/56 D 21/56 E 25/56

ANS: C Use combinations to find numerator and denominator Numerator: Knowing G is a member, how many 3 member teams Denomin: total number of 3 people teams 6C2 = 15 8C3 = 56

If x^m = 1, and m is a non-negative integer, what is the value of x ? (1) x < 0 (2) m is a positive even number

ANS: C (1) x can be any number if m = 0 (2) x could be 1 or -1 Together, x must be -1

If x is not equal to 0, is |x| less than 1? (1) x/|x| < x (2) |x| > x

ANS: C If we know x < 0 (statement 2), we know that x > -1 (statement 1). This means that -1 < x < 0. This means that x is definitely between -1 and 1.

In which quadrant of the coordinate plane does the point (x, y) lie? (1) |xy| + x|y| + |x|y + xy > 0 (2) -x < -y < |y|

ANS: D (1) realize both must be positive otherwise eqn = 0 (2) y is positive if (-x) < (-y) then x > y!! x is also positive

A list contains n distinct integers. Are all n integers consecutive? (1) The average (arithmetic mean) of the list with the lowest number removed is 1 more than the average (arithmetic mean) of the list with the highest number removed. (2) The positive difference between any two numbers in the list is always less than n.

ANS: D (2) Difference = high - low or (n-1)

p and q are different two-digit prime numbers with the same digits, but in reversed order. What is the value of the larger of p and q? (1) p + q = 110 (2) p - q = 36

ANS: D Find valid digits before finding pairs 1 - rule out even because any ones-digit even will be even 2 - rule out 5's 3- p & q cannot be the same {1, 3, 5, 7, 9} 4- only find primes (13, 31), (17, 71), (37, 73) and (79, 97) **there is a "C trap" at work here. The two statements together would allow us to solve for both p and q easily**

Which of the following expressions CANNOT have a negative value? |a + b| - |a - b| |a + b| - |a| |2a + b| - |a + b| a2 + b2 - 2|ab| |a3 + b3| - a - b

ANS: D Rearrange to perfect square a2-2ab-b2 Note that a2 = |a|2

Is x > 0? (1) |x + 3| = 4x - 3 (2) |x + 1| = 2x - 1

ANS: D When there is a variable outside the absolute value, both solutions are not always valid. We need to plug both x = 2 and x = 0 back into the original equation and test them

If x, y, and z are positive, is yz > 2? (1) x + y = 8 (2) z - x = 4

ANS: E Cannot assume x is an integer if x = 7.999, z = .0001 and yz < 2

Five years ago at laboratory B, the ratio of doctorate to nondoctorate researchers was 2:3. If no researchers have resigned, what is the current ratio? (1) In the last five years, twice as many doctorate as nondoctorate researchers were hired. (2) 50 doctorate researchers were hired during the last five years.

ANS: E Need original pool`

In the rectangular coordinate system, points (4, 0) and (- 4, 0) both lie on circle C. What is the maximum possible value of the radius of C ? (A) 2 (B) 4 (C) 8 (D) 16 (e) infinate

ANS: E Takes 3 points to define a circle Depends on where the origin is

For every positive integer n, the nth term of sequence is given by an= 1/n - 1/n+1. What is the sum of the first 100 terms? (a) 1 (b) 0 (c) 25 (d) 99/100 (e) 100/101

ANS: E gut check the denominator! if n = 100, n+1 = 101 Denominator must = 101 Or line up terms a1 = 1/1 - 1/2 a2 = 1/2 - 1/3 .... a99 = 1/99 - 1/100 a100 = 1/100 - 1/101 The middle terms cancel out leavin 1-1/101

If the positive integer N is a perfect square, which of the following must be true? I. The number of distinct factors of N is odd. II. The sum of the distinct factors of N is odd. III. The number of distinct prime factors of N is even.

ANS: I and II I is for sure true II may or may not be true but test numbers

Which of the following triples of numbers have the same standard deviation as the numbers r, s, and t? I. r-2, s-2, t-2 II. 0, r-s, t-s III. r-4, s+5, t-1

ANS: I and II If we add or subtract a constant to each term in a set the standard deviation will not change. II subtracts the constant s

If a, b, c, and d are integers and ab^2c^3d^4 > 0, which of the following must be positive? I. a^2cd II. bc^4d III. a^3c^3d^2

ANS: III only We only know that a and c are the same signs

If a + 1 = 20/a and b is the average of a set of c consecutive integers, where c is odd, which of the following must be true? I. a^2b^2c^2 is even. II. a + b + c is odd. III. ab( c^2 + c) is even.

ANS: III only a+1 = 20/a cannot assume just odd Can also be negative solve for a a^2 + a - 20 = 0

If ab ≠ 0 and a^3b = ab^3, then which of the following must be true? I a = b II a = -b III ab = 1

ANS: None |a| = |b| must be true

RW Press, the advertising rate, a, is inversely proportional to the productivity measure, b, which, in turn, is inversely proportional to the labor cost, c. Is the labor cost at least 200 when the productivity measure is at least 100? (1) When c > 100 , b > 50 (2) When c = 100, a =100

ANS: a inverse proportions can also be written as c1b1 = c2b2

B and C are constants in the equation x^2 + bx + c what is the product of the two roots (which are different)? (1) one root is 3 (2) c = 6 Let q and p be the roots

ANS: b We know that (x+q)(x+p) = provided eqn c = pq as long as you know c, you know pq (doesn't matter the actual values)

The temperature inside a certain industrial machine at time t seconds after startup, for 0 < t < 10, is given by h(t) = 4^(2t) + 1 - 4^(t) + 2 degrees Celsius. How many seconds after startup is the temperature inside the machine equal to 128 degrees Celsius?

ANS: test numbers! 3/2

(0.99999999)/(1.0001)- (0.99999991)/(1.0003) =

Ans: 2(10^-4) Step 1: write using powers of 10 .0001 = 1 x (10^-4) (1-10^-8)/(1+10^-4) Recognize quadratic - expand into (1+10^-8)(1-10^-8), terms cancel out!

For the positive integers q, r, s, and t, the remainder when q is divided by r is 7 and the remainder when s is divided by t is 3. All of the following are possible values for the product rt EXCEPT

Ans: 38 r > 7 and t >3 So answer must be a multiple of factors greater than 4 and 8 38 is divisible by 2 and 19, which are prime

The integer x is positive. What is the remainder when the x is divided by 14? (1) The remainder when 4x is divided by 28 is 12. (2) The remainder when x is divided by 21 is 3.

Ans: E 1 - convert into equations (1) 4x = N28 + 12 Can simplify into x = 7N + 3 2 - test both even and odd "n" numbers can be either remainder 3 or 10 (2) X= 21T + 3 test both even and odd - can be remainder 3 or 10 if remainders only overlap once, i.e. 3, then both together are sufficient

If ¡n! = (n!) 2, then ¡17! - ¡16! = ¡1! (¡16!)(¡4!)(2) (¡16!)(12)(2) 172 (¡16!)(12^2)(2)

Ans: E ¡17! - ¡16! = ¡16! (17×17 - 1) 17 x 17 = 289 -1 = 288

Is n2 > n3? (1) n2 < 1 (2) n3 < 1

Ans: E (1) n can be a positive or negative fraction or 0 If 0, untrue if positive fraction, untrue If negative fraction, true (2) n can be 0 or fraction (negative or postiive) or negative integer if 0, untrue if negative number, true Together n can be 0 or fraction

Alan and Betty live in a multi-story apartment building. How many stories does the building have? (1) There are 3 stories between those on which Alan and Betty live. (2) There are 9 stories above Alan's floor and 9 stories below Betty's floor.

Ans: e Need to know who is above

If the probability that Stock A will increase in value during the next month is 0.54, and the probability that Stock B will increase in value during the next month is 0.68. What is the greatest value for the probability that neither of these two events will occur? A. 0.22 B. 0.32 C. 0.37 D. 0.46 E. 0.63

Cannot assume independent events ANS: B Assume total overlap with A

If 5^x - 5^(x- 3) = 124*5^y, what is y in terms of x?

Factor out 124 (5^3 - 1) Figure out how to make the other side the same Factor out 5^(x-3) y=x-3

If x is the product of the integers from 1 to 150, inclusive, and 5^y is a factor of x, what is the greatest possible value of y ?

Find the power of 5 in the prime factorization of 150 1. Find number of multiples of 5 between 1 and 150. 150 / 5 = 30 2. Find number of multiples of 25 (where 5 occurs twice) between 1 and 150. 150/25 = 6 3. Find number of multiples of 125 = 1 30 + 6+1=37

x^2 - y^2 = ?

If either (x-y) or (x+y) = 0, whole eqn is 0

K and N are integers; k/n remainder 11. if K/N = 81.2 what is n?

K = nt + 11 K = 81n + 11 K = 81.2n 81.2n = 81+11 .2n = 1 n = 55

The set S has 36 different subsets each of which contains exactly two elements. How many subsets of S could contain exactly three elements each?

Permutation question ANS: 84

Does the sequence {a1, a2, a3, ..., an, ...} contain an infinite number of terms that are divisible by 20? (1) a1 = 5 and an = 4(5^(n - 1)) for all integers n ≥ 2. (2) a2 = 20, a4 = 500, a5 = 2,500, and a6 = 12,500.

Trick question, ans: A Don't assume if given defined terms, the remainder will follow a certain pattern

What is the value of the two-digit positive integer n? (1) When n is divided by 5, the remainder is equal to the tens digit of n. (2) When n is divided by 9, the remainder is equal to the tens digit of n.

ans: c recognize patterns (1) write out possibilities by tens When tens =1, ones digit is 1 or 6 when tens = 2, ones digit is 2 or 7 etc (2) same methodology Find the overlap

Determinant

b^2 - 4ac