Задачі для практики

A perfect number is dened as one for which the sum of all the distinct factors less the number itself equals to the number. For instance, 6 is a perfect number, because the factors of 6 (apart from 6 itself) are 1, 2 and 3, and 1+2+3 = 6. Which (A) 12 (B) 28 (C) 15 (D) 13 (E) 67 of the following is a perfect number?

A 12: The required factors are 1, 2, 3, 4, and 6, Their sum = 1 + 2 + 3 + 4 + 6 = 16 ≠ 12. Incorrect.B 20: The required factors are 1, 2, 4, 5, and 10, Their sum = 1 + 2 + 4 + 5 + 10 = 22 ≠ 20. Incorrect.C 28: The required factors are 1, 2, 4, 7, and 14, Their sum = 1 + 2 + 4 + 7 + 14 = 28 = 28. Correct.

How many integers from 1 to 200, inclusive, are divisible by 3 but not divisible by 7? (A) 38 (B) 57 (C) 58 (D) 60 (E) 66

1) Find number of multiples of 3: n=(largest multiple - smallest multiple)/3 + 1 2) Find same for 21 (7*3) 3) відніми числа - буде відповідь! B

(GC E) What is X? (1) X2 − 1 = X + 1 (2) X + 3 is a prime number

1. x^2 - 1 = x + 1x^2 - x - 2 = 0(x-2)(x+1) = 0x = 2 or -1 (Insufficient) 2. x + 3 is a prime numberx could be -1,0,2 and so on. (Insufficient)

What is the sum of digits of number 10^28 − 28?

100=10^21000=10^3.so 100-28=72 . so sum= 91000-28=972. sum= 9+9so 10^28= 9+9+9......27times( i.e 28 -1 times)= 26*9=243

In an arithmetic sequence, if a5 = 16 and d = an − an−1, what is the value of d? (1) a1 = d2 + 4(2) d > 0.

Answer: C

Let N be the greatest integer with distinct digits and multiple of 8 as well. What will be the reminder when N divided by 1000?

Find the biggest number that will be divisible by 8 (3 last digits). The reminder when we divide by 10 (last digit), by 100 (last two digits), and by 1000 (last 3 digits). Answer 120.

A lter decreases concentration of harmful substances in the water by 50%. How many lters are required to clean 200 liters of water containing 0.5 liters of harmful substances if the water that contains less than 0.1 percent of harmful substances is considered clean? (A) 8 (B) 6 (C) 5 (D) 4 (E) 2

First calculate the concentration of the contaminated water:0.5l/200l * 100 = 0.25%Every filter then reduces the concentration by half:The first filter will bring the concentration down to 0.125%. A second one will bring it down to 0.0625%. A concentration that is below the 0.1% that is considered save. Answer A is correct.

What is the number of distinct positive factors of the product of 484 and 124? (A) 3 (B) 4 (C) 7 (D) 8 (E) 30

First we want to find the prime factors of each number:484 = 2^2 * 11^2124 = 2^2 * 31Therefore their product:484*124 = 2^4 * 11^2 * 31Now to find the distinct number of factors we take the the powers of each distinct prime number add one to it and multiply them together:(4+1) * (2+1) * (1+1) = 30 Answer: E

John deposited $10,000 to open a new savings account that earned 4 percent annual interest,compounded quarterly. If there were no other transactions in the account, what was the amount of money in John's account 6 months after the account was opened? (A) $10,100 (B) $10,101 (C) $10,200 (D) $10,201 (E) $10,400

For the first 3 moths interest was 1% of $10,000, so $100;For the next 3 moths interest was 1% of $10,000, plus 1% earned on previous interest of $100, so $100+$1=$101;Total interest for 6 months was $100+$101=$201, hence balance after 6 months was $10,000+ $201=$10,201.Answer: D.

If n is a prime number greater than 3, what is the remainder when n^2 is divided by 12 ? (A) 0 (B) 1 (C) 2 (D) 3 (E) 5

There are several algebraic ways to solve this question, but the easiest way is as follows: since we cannot have two correct answers just pick a prime greater than 3, square it and see what would be the remainder upon division of it by 12.n=5 --> n^2=25 --> remainder upon division 25 by 12 is 1.

If M and N are positive integers, is M divisible by N? (1) Greatest common divisor of M and N is 1 (2) Least common multiple of M and N is 210

There are several values of M and N that satisfy BOTH statements. Here are two: Case a: M = 1 and N = 210. In this case, the answer to the target question is M is not divisible by NCase b: M = 210 and N = 1. In this case, the answer to the target question is M is divisible by NSince we can't answer the target question with certainty, the combined statements are NOT SUFFICIENT Answer: E

If x is a sum of all even integers on the interval 13...65 and y is their number, what is the greatest common divisor of x and y (A) 1 (B) 13 (C) 26 (D) 52 (E) 1014

рахуємо к-сть елементів і запамятовуємо, що в формулі суми фігурує вже к-ість елементів як множник, тому к-сть елементів і є найбільший дільник.

If x2 − 100 < 300, how many integers x satisfy this condition? (A) 42 (B) 39 (C) 38 (D) 37 (E) 19

тут квадрат, тому враховуємо і відємні значення C

What is the range of 10 numbers 2,3,4,6,7,9,10,12,20 and x? (1) The average of the 10 numbers is 8. (2) x is equal to the median of the 10 numbers

якщо х=медіані, значить х розміщений в центрі! Не буде в кінцях послідовності! Answer:D

If median of 5 different positive integers is 20, which of the following is the least possible sum of these 5 integers? (A) 62 (B) 66 (C) 69 (D) 72 (E) 100

1 2 20 21 22 Answer: 66 B

If x is a positive integer ,is √ x>2.5x−5? (1) x < 3 (2) x is a prime number

√x will always be greater than 0

OG) If r and s are positive integers, r is what percent of s? (1) r = 3/4s (2) r + s = 75/100

A

Alex deposited x dollars into a new account that earned 8 percent annual interest, compounded annually. One year later Alex deposited an additional x dollars into the account. If there were no other transactions and if the account contained w dollars at the end of two years, which of the following expresses x in terms of w?

Account at the end of the first year would be 1.08x dollars. At this time x dollars was deposited, hence the account at the beginning of the second year would be (1.08x+x) dollars. Account at the end of the second year would be (1.08x+x)∗1.08=w(1.08x+x)∗1.08=w --> x(1.082+1.08)=wx(1.082+1.08)=w --> x=w(1.08+1.082)x=w(1.08+1.082).Answer: D.

In Company X, 30 percent of the employees live over ten miles from work and 60 percent of the employees who live over ten miles from work are in car pools. If 40 percent of the employees of Company X are in car pools, what percent of the employees of Company X live ten miles or less from work and are in car pools? (A) 12% (B) 20% (C) 22% (D) 28% (E) 32%

C

If 1×2×3× . . . ×n is divisible by 990, what is the least possible value of n?

Factorize and the biggest factor 11 is a correct answer!

What is the largest positive integer n for which 20! is a multiple of 2n ?

Power 2^1 - 20/2 = 10Power 2^2 - 20/4 = 5Power 2^3 - 20/8 = 2Power 2^4 - 20/16 = 1Power 2^5 is 32 that is greater than 20. So the answer is 18

How many prime factors does positive integer n have?(1) n/7 has only one prime factor.(2) 3*n^2 has two different prime factors.

#1 - n/7 has only 1 prime factor. 1: n = 49: 49/7 = 7. 1 prime factor, n has 1 prime factor2: n = 21: 21/7 = 3. 1 prime factor, n has 2 prime factorsInsufficient#2 - 3*n^2 - has 2 different prime factors1: n = 2: 3*4 = 12, 2 and 3 are 2 different prime factors, n has 1 prime factor2: n = 6: 3*6*6 = 3*3*2*6. 2 and 3 - 2 different prime factors, n has 2 prime factors (2 and 3)Insufficienttheir combination doesn't yield any result either.from #1 we get n = 7k where k = prime, so 3∗7∗7∗k23∗7∗7∗k2k can be 3 or 7, which would give us n = 21 and 49, which have 2 and 1 prime factors respectively

What is the greatest common divisor of a + b and 4? (1) Greatest common divisor of a and 4 is 2(2) Greatest common divisor of b and 4 is 2

(1) GCD (a,4) = 2. a is a multiple of 2, but not 4, thus a = 2*odd. No info about b. Not sufficient.(2) GCD (b,4) = 2. b is a multiple of 2, but not 4, thus b = 2*odd. No info about a. Not sufficient.(1)+(2) a + b = 2*odd + 2*odd = 2(odd + odd) = 2*even = {a multiple of 4}. GCD of {a multiple of 4} and 4 is 4. Sufficient.Answer: C.

If x and y are positive integers such that x = 8y + 12, what is the greatest common divisor of x and y? (1) x = 12u, where u is an integer (2) y = 12z, where z is an integer

(1) x = 12u, where u is an integer.Substituting x = 12u into x = 8y + 12 gives 12u = 8y + 12, which leads us to 3(u-1) = 2y. The only thing we know from this is that 3 is a factor of y. Is it GCD of x and y? Not clear: if x = 36, then y = 3 and GCD(x, y) = 3 but if x = 60, then y = 6 and GCD(x, y) = 6. Two different answers. Not sufficient.(2) y = 12z, where z is an integer.Substituting y = 12z into x = 8y + 12 gives x = 8*12z + 12, which leads us to x = 12(8z + 1). So, we have y = 12z and x = 12(8z + 1). Now, as z and 8z + 1 do not share any common factor but 1 (8z and 8z+1 are consecutive integers and consecutive integers do not share any common factor other than 1. As 8z has all the factors of z, then z and 8z+1 also do not share any common factor other than 1), then 12 must be the GCD of x and y. Sufficient.Answer: B.

p, ps, ps2, ps3, ps4, ps5 progression. The sum of the first three terms in the series is one-ninth the sum of the first six terms. What is the ratio of the 5th term to the second term? (A) 1:1 (B) 2:1 (C) 3:1 (D) 8:1 (E) 9:1

(p+ps+ps^2)/(p+ps+ps^2+ps^3+ps^4+ps^5)= 1/9 (1/27+1/27+1/27)/(1/27+1/27+1/27+8/27+8/27+8/27) -> (8/27)/ (1/27) 8 to 1 Answer: D

What is the sum of all positive multiples of 6 that are less than or equal to 714? (A) 3,600(B) 7,200(C) 14,000 (D) 28,800 (E) 42,840

1) find n= ((an-a1)/d)+1 2)find sum Answer: E

What is the greatest common divisor of a, b, and c? (1) greatest common divisor of a and b is 3(2) greatest common divisor of b and c is 4

1) ns 2) ns together we can say that they are mutually prime

X,Y,andZ are positive integers. Is(X−Y)×(Y −Z)×(X−Z)>0? (1) X^2 + Y × Z = X × Y + X × Z (2)X×Y −Y2 =X×Z−Y ×Z

1) solve the equation and you get (x-z)(x-y)=0 Which means either of the terms are zero. Using this inference, (x-y)(y-z)(x-z) = 0. Which is not greater than 0. Sufficient 2) similarly st. 2 gives (x-y)(y-z)=0Thus sufficient Answer-D

What is the greatest common divisor of a, b, and 3a + 23b? (1) a = 4 (2) The greatest common divisor of a, b, and 23a + 3b is 4.

1) розписати за формулою і цього НС 2) тут ідуть ті ж самі числа, а тому відповідь В

S = 0.abc, where a, b, and c are any decimal digits. Is S > 2/3? (1) a + b > 14 (2) a + c > 15

23=0.66666...23=0.66666.... We need to know a first. If a≥7 or a≤5, then this is sufficient. If a=6, then we need to look into b and c.Statement 1 Alone:The maximum value for b is 9, so we must have a>5 which is a≥6.Thus when a=6, we have b=9, then S > 2323. When a≥7, we also must have S > 2323. Then this is sufficient.Statement 2 Alone:The maximum value for c is 9, so we must have a>6 which is a≥b As mentioned above, this is sufficient. Answer: D

Find the least positive number, which leaves, if divided by 3, 5, and 12, a remainder of 2. (A) 62 (B) 42 (C) 12 (D) 2 (E) 122

2=3*0+2 2=5*0+2 2=12*0+2 Answer: D

If the product of the integers from 1 to n is divisible by 490, what is the least possible value of n? (A) 7 (B) 14 (C) 21 (D) 28 (E) 35

Answer: B

Is integer x divisible by 3? (1) When x is divided by 5, the remainder is 1. (2) When x is divided by 15, the remainder is 1.

Analysing Statement 1 • As per the information given in statement 1, when x is divided by 5 the remainder is 1o From this statement, we can say different possible values of x can be 6, 11, 16, 21 etco Values like 6 and 21 are divisible by 3, whereas values like 11 and 16 are not divisible by 3 Hence, statement 1 is not sufficient to answerAnalysing Statement 2 • As per the information given in statement 2, when x is divided by 15 the remainder is 1o As the number 15 itself is a multiple of 3, the remainder will be same when x is divided by either 15 or 3o Therefore, we can say the integer x is not divisible by 3 Hence, statement 2 is sufficient to answerHence, the correct answer is option B.

If xn = 1, where x and n are both integers, what is the value of the x? (1) n is a multiple of 5.(2) n is an odd number.

Answer B

Are the integers a, b, and c consecutive? (1) b − a = c − b (2) The average of a, b, and c equals to b

Answer E 1) Try to put: a=0, b=2, c=4 (not consecutive) 2) try same here - different values

OG) If 1 < d < 2, is the tenths' digit of the decimal representation of d equal to 9? (1) d + 0.01 < 2 (2) d + 0.05 > 2

Answer is B. Tenths' digits goes right after dot

If a is a positive integer, and if the units' digit of a2 is 9, and the units' digit of (a+1)2 is 4, what is the units' digit of (a+2)2? (A) 1 (B) 3 (C) 5 (D) 7 (E) 9

Answer: A

What is the greatest common divisor of positive integers x and y? (1) x and y share exactly one common factor (2) x and y are both prime numbers

Answer: A

If 60 students took a test and the median score was 80, which of the following must be true? I. At least 29 scores are greater than 80II. At least 30 scores are equal to or greater than 80 III. At most 30 scores are equal to or less than 80 (A) I only (B) II only (C) III only (D) I and II only (E) I and III only

Answer: B

If p and q are prime numbers, where p is no more than q,is sum of p and q a prime number? (1) p is not equal to q. (2) p is greater than 2.

Answer: B

How many numbers less than 500 exist such that when any of these is divided by 7 the remainder is 3 and when any of these is divided by 3 the remainder is 1?

Explanation:x = 7a + 3x = 3b + 1First digit which gives remainder 3 when divided by 7 and remainder 1 when divided by 3 is 10.x = 21c + 10x must be less than 500, so21c + 10 < 50021c < 490c < 23.3As, c can be any integer between 0 and 23, so total = 24 values

500 records have a distribution that is symmetric about the mean which is 10. If 68 percent of the distribution lie within one standard deviation of the mean, and the standard deviation is 2, then how many of the records are less than 12? (A) 160 (B) 320 (C) 340 (D) 420 (E) 460

If the distribution is symmetric, the number of values below the mean is equal to the number above the mean. If none of the values equal the mean exactly (a technicality we don't need to worry about), we have 250 'below average' records, i.e. with values less than 10. We also know that 68% lie within one standard deviation - 34% of these will be below average, 34% above, because the distribution is symmetric. So from 10 to 12, we have 34% of our values, and below 10 we have another 50%, meaning 84% will have values less than 12. 84% of 500 is 420.

How many positive integers, from 2 to 100, inclusive, are not divisible by odd integers greater than 1? (A) 5 (B) 6 (C) 8 (D) 10 (E) 50

Integer not to have an odd factor greater than 1 should be of the type 2^n, for some positive integer n. So we are given that 2≤2n≤1002≤2^n≤100 --> n can take the following values: 1, 2, 3, 4, 5, and 6, which means that the number itself could be 2, 4, 8, 16, 32, or 64. Answer: B.

On the number line segment [0, 1] has been divided by tick marks into fths and into sevenths What is the least possible distance between any two of the tick marks?

LCM (5,7)= 35 - total distance between (0 and 1). Then for 7: 7-14-21-28-35, and for 5: 5-10-15-20-25-30-35. The least distance is from 14 to 15 or 20 to 21. So 1/35 is the least distance

On a certain road, 10 percent of the motorists exceed the posted speed limit and receive speeding tickets, but 20 percent of the motorists who exceed the posted speed limit do not receive speeding tickets. What percent of the motorists on that road exceed the posted speed limit? (A) 10.5% (B) 12.5% (C) 15% (D) 22% (E) 30%

On a certain road, 10 percent of the motorists exceed the posted speed limit and receive speeding tickets,>>>>10% of total motorists exceed speed limit and get ticketsbut 20% of the motorists who exceed the posted speed limit do not receive speeding tickets. >>>>>>20% who exceed the speed limit do not get tickets. Then 80% who exceed the speed limit must be getting tickets.What percent of the motorists on that road exceed the posted speed limit?>>>>> So 10% of total motorists = 80% of those who exceed speed limitSo those who exceed speed limits are 1/8 = 12.5% of the motorists

$20,000 was deposited into bank, with interest compounded quarterly. What is compound interest? (1) The interest in the second quarter is 1.1 times more than the rst quarter. (2) The interest in the second quarter is 200 dollars more than the rst quarter.

Statement 1- This statement tells us that Interest in the second quarter we get on the interest of the first quarter,x, is 0.1x.Our interest rate is 0.1xx∗100=100.1xx∗100=10SufficientStatement 2- The CI of 1st quarter = 20000∗r100=200r20000∗r100=200r200r∗r100=200200r∗r100=200r^2 = 100r=10%Sufficient

The sum of the fourth and twelfth term of an arithmetic progression is 20. What is the sum of the rst 15 terms of the arithmetic progression? A) 300 (B) 120 (C) 150 (D) 170 (E) 270

a4= a1 +3d a12= a1 +11d S15 = (a1+(a1+14d)/2)*n= 20/2 *15= 150

A person travels in a car with uniform speed. He observes a mile-stone, which has 2 digits. After one hour he observes another mile-stone with same digits reversed. After another hour he observes another mile-stone with same 2 digits containing a 0. Find the speed of the car in miles per hour, if mile-stones tell you how far one has travelled from the starting point. A) 40 (B) 45 (C) 50 (D) 55 (E) 60

ab - decimal value 10a+b ba - decimal value 10b+a Distance he covered from the first to second check-point: 10b+a - 10a -b= 9b-9a - speed should be divisible by 9! Answer:B

If the circus were to sell all of its 220 tickets for this month's performance at their usual price, the revenue from sales would be 10% greater than that collected last month. However, the circus raised the ticket price by 5% and sold only 200 tickets as a result. What percent less was last month's revenue than that of this month?

et the usual price of a ticket be $20 (I chose $20 because $20+5%=$21=integer, which will make calculations easier). This month's revenue for this price would be 220*$20=$4,400 and we are told that it's 10% greater than the revenue collected last month, hence the last month's revenue was $4,400/1.1=$4,000;Circus raised the ticket price by 5%, so the new price was $21 and the actual revenue from 200 tickets was 200*$21=$4,200;What percent less would last month's revenue be compared to this month's revenue: percent=4,200−4,0004,200∗100=10021percent=4,200−4,0004,200∗100=10021. General formula for percent increase or decrease, (percent change): percent=ChangeOriginal∗100percent=ChangeOriginal∗100, so as we are comparing the difference to this month's revenue we should put this value ($4,200) in the denominator.

If m = pxqy, where p and q are dierent positive prime numbers, is m a square of integer? (1) xy is an odd number(2) x + y is àn even number.

for a perfect square the factors of the no will always be odd so #1 given x*y is odd no. means factors of m will be even so its not perfect square no#2x+y is an even no x & y can be odd so yes and x&y are even so noIMO A sufficient

Is x2 −5x+6=|x−2|×|x−3|? (1) x > 2. (2) x > 3.

https://gmatclub.com/forum/is-x-2-5x-6-x-2-x-320032.html

When positive integer n is divided by 13, the remainder is 2 and the quotient is k. When n is divided by 17, the remainder is 2. What is the remainder when k is divided by 17?(A) 0(B) 2(C) 4(D) 13(E) 15

n= 13k+2n= 17x+2then 13k= 17xk must be multiple of 17 because 13 and 17 both are prime.Reminder when dividing k by 17 is 0.A is answer

Is the range of a non-empty set S greater than its mean? (1) All members of S are negative(2) The median of set S is negative

negative mean is always less than range, as range cannot be negative

A, B, C are digits where A×B is not zero. If AB, BA are both two-digit numbers and AAC is a three-digit number, what is the value of B? (1) AB+BA = AAC (2) A=1

методом підстановки!

13. (GC) The price of a certain commodity increased at a rate of X% per year between 2000 and 2004. If the price was A dollars in 2001 and B dollars in 2003, what was the price in 2002 in terms of A and B?

price in 2000 = Aprice in 2001 = A*(1 + X/100)price in 2002 = A*(1 + X/100)^2price in 2003 = A*(1 + X/100)^3price in 2004 = A*(1 + X/100)^4For simplicity, I am going to define r = (1 + X/100). Then these equations become:price in 2000 = Aprice in 2001 = A*rprice in 2002 = A*r^2price in 2003 = A*r^3price in 2004 = A*r^4Now, suppose we have M = 2001 price = A*r and N = 2003 price = A*r^3. How do we represent the 2002 prince (A*r^2) in terms of M and N? express r in terms of M and NThis is more a crank-it-out algebraic solution approach. We notice that N/M = (A*r^3)/(A*r) = r^2, so r = sqrt(N/M). Well,2002 price = (2001 price)*(r) = M * sqrt(N/M) = [sqrt(M)*sqrt(M)]*[sqrt(N)/sqrt(M)] = sqrt(M)*sqrt(N) = sqrt(NM).

How many integers between 100 and 150 inclusive can be evenly divided by neither 3 nor 5? (A) 22 (B) 24 (C) 26 (D) 27 (E) 28

should remove overlap Answer: D

If n = 4p, where p is a prime number greater than 2, how many different positive even divisors does n have, including n ? (A) Two (B) Three (C) Four (D) Six (E) Eight

since p is prime greater than 2, then p=odd, thus 4p=even, which means that it has 4 even divisors: 2, 4, 2p, and 4p. Answer: C

For numbers x, y, z and w, if x is 50% greater than y, and z is 20% greater than w, x is what percent greater or less than z ? (A) 20 % greater (B) 20 % less (C) 25 % greater (D) 25 % less (E) 50 % greater

w=100 x=150 y=100 z=1201. x/y = 150/100 = 3/22. z / w = 120 /100 = 6/5Difference = 3/2 -6/5 = 3/10Required % = (3/10 ) / (6/5)= 1/ 4 ~ 25%.

Ifx2 +12x+20<0,howmanyintegervaluescanxbe? (A) 6 (B) 7 (C) 8 (D) 9 (E) 10

x^2 + 12x + 20 < 0 --> (x+10)(x+2)<0 this will be true only when x<-2 and x> -10.Ans = C

The trinomial x2 + x − 20 is exactly divisible by (A) x − 5 (B) x + 4 (C) x−10 (D) x − 4 (E) x − 2

x^2+bx+c= (x-x1)(x-x2)

A cashier mentally reversed the digits of one customer's correct amount of change and thus gave the customer an incorrect amount of change. If the cash register contained 45 cents more than it should have as a result of this error, which of the following could have been the correct amount of change in cents? (A) 14 (B) 45 (C) 54 (D) 65 (E) 83

xy-right change, yx-reversed! Need remember the rule - we can write numbers like that (digit form) 10x+y; 10y+x. Form the equation: 10x+y -10y-x=45 9x-9y=45 x-y=5 - difference between digits should be 5! Answer: E

What is the greatest possible common divisor of two dierent positive integers which are less than 144? (A) 143 (B) 71 (C) 72 (D) 13 (E) 11

беремо число 143 ділимо на 3 буде 71, а отже 71 тут і буде відповідю