Which of the following CANNOT be expressed as the sum of two or more consecutive positive integers?

Which of the following CANNOT be expressed as the sum of two or more consecutive positive integers?

(A) 17

(B) 22

(C) 24

(D) 26

(E) 32

 

6. If m is an integer, which of the following could be true?

I.  is an even integer.

II.  is an even integer.

III. 17m is a prime.

(A) I only

(B) II only

(C) III only

(D) I and II only

(E) II and III only

 

 

7. The number p is greater than 0, a multiple of  6, and a factor of 180. How many possibilities are there for the value of  p?

a. 7

b. 8

c. 9

d. 10

e. 11

 

 

8. For any whole number x > 0, how many elements are in the set that contains only the numbers that are multiples AND factors of x?

 

 

9. If n is a positive integer, which of the following is always odd?

(A) 19n + 6

(B) 19n + 5

(C) 19n² + 5

(D) 18n + 4

(E) 18n + 5

 

 

X is divisible by 3 and 5 so what is x ?

 

11. 134xy is divisible by 5 and 9. What may be x y ?

 

If the sum of seven integers is even, at most how many of these integers could be odd?

(A) 3

(B) 4

(C) 5

(D) 6

(E) 7

 

 

13. If n is an integer, which of the following CANNOT be a factor of 3n + 4?

(A) 4

(B) 5

(C) 6

(D) 7

(E) 8

 

 

X is an even number, which of the following is odd?

(a) X².

(b) (X+1)²

(c) (X+2)²

(d) X³ + X

(e) 2X²

 

15. A is a prime number (A > 2). If С = A³ then in how many different integers is С divisible?

a) 5

b) 6

c) 3

d) 4

e) 7

 

 

Eggs are sold in packages of six or eleven only. If Doris bought 70 eggs exactly, what could be the number of large packs Doris bought?

a) 6.

b) 2.

c) 3.

d) 5.

e) 4.

 

 

17. If n is the greatest positive integer for which 2 n is a factor of 8!, then n =

A. 2

B. 4

C. 6

D. 7

E. 10

 

 

18. If 10⁵⁰ – 74 is written as an integer in base 10 notation, what is the sum of the digits in

That integer?

A. 424

B. 433

C. 440

D. 449

E. 467

 

19. If n = 8⁵ – 8, what is the units digit of n?

A. 0

B. 1

C. 4

D. 6

E. 8

 

20. If y is the smallest positive integer such that 3,150 multiplied by y is the square of an integer, then y must be

A. 2

B. 5

C. 6

D. 7

E. 14

 

21. When a positive integer, m , is divided by 7, the remainder is 4. Which of the following expressions will yield a remainder of 3 when it is divided by 7?

(A) m + 2

(B) m + 3

(C) m + 4

(D) m + 5

(E) m + 6

 

22. If n = 5²⁰⁰⁰ + 5²⁰⁰², then what are the prime factors of n?

(A) 5 only

(B) 2 and 5 only

(C) 2, 5, and 10 only

(D) 2, 5, and 13 only

(E) 2, 5, 1000, and 1001 only

 

 

23. If a is a multiple of 7 and b is a prime number bigger than 5, then which of the following must be an odd integer?

(A) a/b

(B) ab

(C) a + b

(D) 2a + b

(E) a + 2b

 

 

24. If x is the product of the positive integers from 1 to 8, inclusive, and if i , k , m , and p are positive integers such that x = 2 ⁱ 3 ^k 5 ^m 7 ^p , then i + k + m + p =

A. 4

B. 7

C. 8

D. 11

E. 12

 

 

25. If n is a positive integer, what is the remainder when  6ⁿ is divided by 10 ?

A. 1

B. 2

C. 4

D. 6

E. 8

 

 

Random two-digit integers can be generated using ten cards numbered from zero to nine. A card is drawn at random from the ten cards, and that number is recorded as the first digit of the two-digit integer. The card is replaced, and a card is again drawn at random from the ten cards. This number is recorded as the second digit or the two-digit integer. If the first card drawn is a 2, how many prime two-digit integers can result?

A. 2

B. 3

C. 4

D. 5

E. 6

 

36. When the four numbers  and  are arranged in order from smallest to largest, the result is  and . Which of the following is a possible value for b?

A.

B.

C.

D.

E.

 

37. A certain number y has a remainder of 3 when divided by 5. That same number has a remainder of 2 when divided by 4. If y is an integer between 10 and 40, what is one possible value of y?

 

 

Multiple of 6?

a) Q + 2.

b) Q-4.

c) 2Q + 6.

d) 3Q.

e) 4Q + 2.

 

Which of the following CANNOT be expressed as the sum of two or more consecutive positive integers?

(A) 17

(B) 22

(C) 24

(D) 26

(E) 32

 

6. If m is an integer, which of the following could be true?

I.  is an even integer.

II.  is an even integer.

III. 17m is a prime.

(A) I only

(B) II only

(C) III only

(D) I and II only

(E) II and III only

 

 

7. The number p is greater than 0, a multiple of  6, and a factor of 180. How many possibilities are there for the value of  p?

a. 7

b. 8

c. 9

d. 10

e. 11

 

 

8. For any whole number x > 0, how many elements are in the set that contains only the numbers that are multiples AND factors of x?

 

 

9. If n is a positive integer, which of the following is always odd?

(A) 19n + 6

(B) 19n + 5

(C) 19n² + 5

(D) 18n + 4

(E) 18n + 5

 

 

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