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QUESTION: 1

8 k 8

+k 8 8

-------------------------

1,6p6

If *k* and *p* are non-zero digits within the integers above, what is the value of *p*?

Solution:

QUESTION: 2

When the positive integer x is divided by 11, the quotient is y and the remainder 3. When x is divided by 19, the remainder is also 3. What is the remainder when y is divided by 19?

Solution:

QUESTION: 3

What is the sum of the digits of the positive integer *n* where n < 99?

1) *n* is divisible by the square of the prime number y.?

2) y^{4} is a two-digit odd integer.

Solution:

QUESTION: 4

List K consists of 12 consecutive integers, if -4 is the least integer in list K, what is the range of the positive integers in the list K? ?

Solution:

QUESTION: 5

If *x* represents the sum of all the positive three digit numbers that can be constructed by using each of the distinct nonzero digits a, b, and c exactly once, what is the largest integer by which x must be divisible?

Solution:

QUESTION: 6

If x is the sum of six consecutive integers, then x is divisible by which of the following:

I. 3 II. 4 III. 6

Solution:

QUESTION: 7

In a certain deck of cards, each card has a positive integer written on it. In a multiplication game, a child draws a card and multiplies the integer on the card by the next larger integer. If each possible product is between 15 and 200, then the least and greatest integers on the cards could be?

Solution:

QUESTION: 8

The number 75 can be written as the sum of the squares of 3 different positive integers. What is the sum of these 3 integers?

Solution:

QUESTION: 9

If a six-digit number is constructed using the three digits *a, b, *and *c* in the following manner: *abcabc, *then which of the following would always be a factor of *abcabc*?

Solution:

QUESTION: 10

For a finite sequence 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 two consecutive terms is negative. What is the number of variations is in sign for the sequence: 1, -3, 2, 5, -4, -6? ?

Solution:

QUESTION: 11

Is *z* a perfect square?

1) *z* has exactly one factor

2) *z* is a positive integer

Solution:

QUESTION: 12

If xy + z = x(y + z), which of the following must be true?

Solution:

QUESTION: 13

If q, r, and s are consecutive even integers and q < r < s, which of the following CANNOT be the value of s^{2} – r^{2} – q^{2}? ?

Solution:

QUESTION: 14

Is *x* a perfect square?

1) *x* ≤ 1

2) *x* is a non-negative number

Solution:

QUESTION: 15

The sum of positive integers x and y is 77. What is the value of xy?

- x = y + 1
- x and y have the same tens' digit. ?

Solution:

QUESTION: 16

The sum of positive integers x and y is 75. What is the value of xy?

- x = y + 1
- x and y have the same tens' digit. ?

Solution:

QUESTION: 17

Symbol * denote to be one of the operations add, subtract, multiply, or divide. Is (6*2)*4 = 6*(2*4)?

1) 3*2 > 3

2) 3*1 = 3

Solution:

QUESTION: 18

If m and r are two numbers on a number line, what is the value of r??

1) The distance between r and 0 is 3 times the distance between m and 0.

2) 12 is halfway between m and r

Solution:

QUESTION: 19

If k and x are positive integers and x is divisible by 6, which of the following CANNOT be the value of √(288kx)?

Solution:

QUESTION: 20

*x, y, a,* and *b* are positive integers. When *x* is divided by *y*, the remainder is 6. When *a* is divided by *b*, the remainder is 9. Which of the following is NOT a possible value for *y + b*??

Solution:

QUESTION: 21

In order to play a certain game, 24 players must be split into n teams, with each team having an equal number of players. If there are more than two teams, and if each team has more than two players, how many teams are there? ?

(1) If thirteen new players join the game, one must sit out so that the rest can be split up evenly among the teams. ?

(2) If seven new players join the game, one must sit out so that the rest can be split up evenly among the teams. ?

Solution:

QUESTION: 22

When the integer x is divided by the integer y, the remainder is 60. Which of the following is a possible value of the quotient x/y? ?

I. 15.15 II. 18.16 III. 17.17?

Solution:

QUESTION: 23

Five consecutive positive integers are chosen at random. If the average of the five integers is odd, what is the remainder when the largest of the five integers is divided by 4??

(1) The third of the five integers is a prime number. ?

(2) The second of the five integers is the square of an integer.

Solution:

QUESTION: 24

If a and b are both single-digit positive integers, is a + b a multiple of 3? ?

(1) The two-digit number "ab" (where a is in the tens place and b is in the ones place) is a multiple of 3.

(2) a – 2b is a multiple of 3.

Solution:

QUESTION: 25

Is n a multiple of 3?

1) (n – 1) is a multiple of 2

2) (n + 1) is a multiple of 4

Solution:

QUESTION: 26

Is y a multiple of 3?

1) (y – 1)(y)(y + 1) is a multiple of 3

2) y(y + 1) is a multiple of 3

Solution:

QUESTION: 27

What is the remainder when x is divided by 4?

1) (x + 1) leaves a remainder of 2 when divided by 8

2) x leaves a remainder of 1 when divided by 2

Solution:

QUESTION: 28

When x is divided by 4, the quotient is y and the remainder is 1. When x is divided by 7, the quotient is z and the remainder is 6. Which of the following is the value of y in terms of z??

Solution:

QUESTION: 29

If positive integer n is divisible by both 4 and 21, then n must be divisible by which of the following?

Solution:

QUESTION: 30

If p^{3} is divisible by 80, then the positive integer p must have at least how many distinct factors?

Solution:

QUESTION: 31

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

Solution:

QUESTION: 32

How many integers are there between 51 and 107, inclusive??

Solution:

QUESTION: 33

What is the sum of all the 4 digit integers that can be created using the digits 1, 2, 3, and 4?

Solution:

QUESTION: 34

What is the three-digit number abc, given that a, b, and c are the positive single digits that make up the number?

(1) a = 1.5b and b = 1.5c?

(2) a = 1.5x + b and b = x + c, where x represents a positive single digit

Solution:

QUESTION: 35

If x and y are positive integers, is x > y?

1) √x > y

2) x^{2} > y

Solution:

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