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INTEGERS PROBLEM SET

1) r and s are integers and rs + r isodd, which MUST be even?

a. rb. sc. r + sd. rs re. r2 + s

2) Is integer r divisible by 3?a. r is the product of 4

consecutive positive integers

b. r < 253) n is an integer between 10 and 99,

is n < 80?

a. Sum of the two digits on nis a prime number.

b. Each of the two digits of nis a prime number

4) If n is a positive integer less than200 and 14n60 is also an integer,

then n has how many different

positive prime factors

a. 2b. 3c. 5d. 6e. 8

5) If r and s are positive integers, isr/s an integer?

a. Every factor of s is also afactor of r.

b. Every prime factor of s isalso a prime factor of r.

6) Which of the following is andinteger?

I. 12!/6!II. 12!/8!III. 12!/(7!x5!)

a. Ib. IIc. IIId. I & IIe. I, II & III

7) n and m are positive integers,what is remainder when 34n + 2 + m

is divided by 10?

a. n = 2b. m = 1

8) If x and y are integers greaterthan 1, is x a multiple of y?

a. 3y2 + 7y = xb. x2 x is a multiple of y

9) x, y, and z are integers greaterthan 1, what is value of x + y + z?

a. xyz = 70b. x/yz = 7/10

10)Is z equal to the median of the 3positive integers x,y & z?

a. x < y + zb. y = z

11)n and y are positive integers and450y = n

3, which is an integer?

I. y / 3 * 22 * 5II. y / 32 * 2 * 5III. y / 3 * 2 * 52

a. Noneb. Ic. IId. IIIe. I, II, III

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12)What is the remainder when positiveinteger X is divided by 3?

a. When X is divided by 6remainder is 2.

b. When X is divided by 15,remainder is 2.

13)Sum of positive integers X and Y is72. What is XY?

a. X = Y + 1b. X and Y have same tens digit.

14)If X is negative integer, which of thefollowing has the least value?

a. X2 0.5b. X2 0.51c. X3 0.502d. X3 0.511e. X4 0.512

15)For how many integers N is 2N = N2a. Noneb. 1c. 2d. 3e. More than 3?

16)What is GCD of positive integers Mand N?

a. M is a prime numberb. 2N = 7M

17)If ZN = 1, what is value of Z?a. N is a nonzero integerb. Z > 0

18)Positive integers X, Y, and Z are suchthat X is a factor of Y and Y is a factor

of Z, if Z even?

a. XZ is evenb. Y is even

19)If X is a positive number less than 10,is Z greater than average of X and

10?

a. Z is closer to 10 than X on thenumber line

b. Z = 5X20)For every positive even integer n,

the function H(n) is defined to be the

product of all even integers from 2 to

n inclusive. If p is the smallest

prime factor of H(100) + 1, then p

is between

a. 2 and 10b. 10 and 20c. 20 and 30d. 30 and 40e. > 40

21)75 can be written as the sum of thesquares of 3 different positive

integers. What is sum of these

integers?

a. 17b. 16c.

15

d. 14e. 13

22)If P is positive integer, what isremainder when P is divided by 4?

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a. When P is divided by 8remainder is 5.

b. P is the sum of the squares oftwo consecutive positive

integers.

23)Is M + Z > 0?a. M 3Z > 0b. 4Z M > 0

24)A set of 15 different integers hasmedian of 25 and a range of 25.

What is greatest possible integer

that could be in this set?

a. 32b. 37c. 40d. 43e. 50

25)How many odd integers are greaterthan integer X and less than integer

Y?

a. There are 12 even integersgreater than X and less than

Y.

b. There are 24 integers greaterthan X and less than Y.

26)If n is a positive integer and r isremainder when (n-1)(n+1) is divided

by 24, what is the value of r?

a. n is divisible by 2.

b. n is not divisible by 3.

27)Positive integer K has exactly 2positive prime factors, 3 and 7. If K

has a total of 6 positive factors,

including 1 and K, what is the value

of K?

a. 32 is a factor of K.b. 72 is not a factor of K.

28)If integer N is > 1, is N = 2?a. N has exactly 2 positive

factors.

b. The difference of any 2distinct positive factors of N

is odd.

29)If x is positive, which of the followingcould be the correct ordering of 1/x,

2x, and x2

?

I. x2 < 2x < 1/xII. x2 < 1/x < 2xIII. 2x < x2 < 1/x

a. Noneb. Ic. IIId. I and IIe. I, II, and III

30)If K is a positive integer, then 20K isdivisible by how many different

positive integers?

a. K is prime.

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b. K is 7.31)How many different prime numbers

are factors of positive integer n?

a. 4 different prime numbersare factors of 2n.

b. 4 different prime numbersare factors of n2.

32)X is the sum of even integers from 40to 60 inclusive and Y is the number

of even integers from 40 to 60

inclusive, what is X + Y?

a. 550b. 551c. 560d. 561e. 572

33)If N and P are integers, is P > 0?a. N + 1 > 0b. NP > 0

34)Are positive integers P and Q bothgreater than N?

a. P - Q > Nb. Q > P

35)Sum of 1st 50 positive even integersis 2550. What is sum of even integers

from 102 to 200 inclusive?

a. 5100b. 7550c. 10100d. 15500e. 20100

36)If C and D are integers, is C even?a. C(D +1) is evenb. (C+2)(D+4) is odd

37)If 11 consecutive integers are listedfrom least to most, which is the

mean of the 11 digits?

a. Average of the first 9 digits is7.

b. Average of last 9 digits is 9.38)The function f is defined for all the

integers n by f(n) is the number of

positive integers each of which is less

than n and has no positive factor in

common with n other than 1. If p

is any prime number, then f(p) =?

a. p 1b. p 2c. p+2 / 2d. p-1 / 2e. 2

39)If K and T are integers, and K2 T2 isan odd integer, which must be even?

I. K + T + 2II. K2 + 2kt + T2III. K2 + T2

a. Noneb. Ic. IId. IIIe. I, II, and III

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40)R and T are positive integers, is RTeven?

a. R + T is oddb. RT is odd

41)What is greatest prime factor of 417 228?

a. 2b. 3c. 5d. 7e. 11

42)If X 0, then X2/X =a. -1b. 0c. 1d. Xe. |X|/X

43)If # is defined by a # b = a + b ab,then which is true?

I. a # b = b # aII. a # 0 = aIII. (a # b) # c = a # (b # c)

a. Ib. IIc. I & IId.

I & III

e. I, II & III44)Is X between 0 and 1?

a. X is between -12 and 32b. 34 is 14 more than X

45)If M, R, X & Y are positive, is the ratioof M to R equal to the ratio of X to Y?

a. Ratio of M to Y is equal toratio of X to R

b. Ratio of M + X to R + Y equalsratio of X to Y

46)If the average of five numbers, X, 7,2, 16, and 11 equals the median,

what is X?

a. 7 < X > 11b. X is median of the five

numbers

47)If XYZ > 0, is X > 0?a. XY > 0b. XZ > 0

48)For a finite set of nonzero numbers,the number of variations in sign is

defined as the number of pairs of

consecutive terms of the sequence

for which the product of the

consecutive terms is negative. What

is the number of variations in sign for

the sequence 1, -3, 2, 5, -4, -6?

a. 1b. 2c. 3d. 4e. 5

49)Is X less than 20?a. Sum of X and Y is less than 20b. Y is less than 20

50)What is the value of N?

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a. N is between 0 and 1b. 716 is 38 more than N

51)If XY + Z = X(Y + Z), which must betrue?

a. x = 0; z = 0b. x = 1 or y = 1c. y = 1; z = 0d. x = 1 or y = 0e. x = 1 or z = 0

52)S is finite set of numbers. Does Scontain more negative numbers than

positive numbers?

a. Product of all numbers in S is-1200

b. There are 6 numbers in S53)If X4 + Y4 = 100, the greatest possible

value of X is between

a. 0 and 3b. 3 and 6c. 6 and 9d. 9 and 12e. 12 and 15

54)If each term in the series a1 + a2 + an is either 7 or 77, and the sum

equals 350, which of the following

could be equal to n?

a. 38b. 39c. 40d. 41e. 42

55)If X > Y2 > Z4, which of the followingcould be true?

I. x > y > zII. z > y > xIII. x > z > y

a. Ib. I and IIc. I and IIId. II and IIIe. I, II, and III

56)On the number line, if K is to the leftof T, is KT to the right of T?

a. T < 0b. K < 1

57)If 0 < r < 1 < s < 2 which must be lessthan 1?

I. rsII. rsIII. s r

a. Ib. IIc. IIId. I and IIe. I and III

58)Are X and Y both positive?a. 2X 2Y = 1b. XY > 1

59)If MV < PV < 0, is V > 0?a. M < Pb. M < 0

60)If S and T are two different numberson the number line, is S + T = 0?

a. Distance from S to 0 is thesame as the distance from T

to 0

b. 0 is between S and T