Survey of Mathematics - Exam Solutions

1. If

57873 = q · 10 + r, 0 ≤ r < 10,

then
(A) q = 57873, r = 0
(B) q = 7873, r = 5
(C) q = 5787, r = 3
(D) q = 5787, r = 0
(E) None of these

Answer: (C) q = 5787, r = 3

2. We wish to find q and r such that

176533 = q · 311 + r, 0 ≤ r < 311

The calculator says that 176533 ÷ 311 567.6302. Therefore

(A) q = 567, r = 0
(B) q = 567, r = 100
(C) q = 568, r = 196
(D) q = 567, r = 196
(E) None of these

Answer: (D) q = 567, r = 196

3. The greatest common factor of 105 and 75 is

(A) 3
(B) 5
(C) 9
(D) 15
(E) None of these

Answer: (D) 15

4. If gcf (105, 45) = 105u + 45v, then

(A) u = 1, v = −2
(B) u = 2, v = −1
(C) u = 1, v = 2
(D) u = 2, v = −2
(E) None of these

Answer: (A) u = 1, v = −2

5. The greatest common factor of 385 and 15 is

(A) 3
(B) 5
(C) 9
(D) 15
(E) None of these

Answer: (B) 5

6. If gcf (385, 15) = 385u + 15v, then

(A) u = 1, v = −25
(B) u = −1, v = 25
(C) u = 1, v = −26
(D) u = −1, v = 26
(E) None of these

Answer: (D) u = −1, v = 26

7. The greatest common factor of 4269 and 2134 is

(A) 1
(B) 3
(C) 7
(D) 11
(E) None of these

Answer: (A) 1

8.If gcf (4269, 2134) = 4269u + 2134v, then

(A) u = 1, v = −2
(B) u = −3, v = 5
(C) u = 1, v = −3
(D) u = −3, v = 4
(E) None of these

Answer: (A) u = 1, v = −2

9. The least common multiple of 15 and 35 is

(A) 30
(B) 35
(C) 45
(D) 70
(E) None of these

Answer: (E) lcm (15, 35) = 105

10. If ab = 79, 200 and gcf(a, b) = 30, then lcm(a, b) =

(A) 79200
(B) 26400
(C) 7920
(D) 2640
(E) None of these

Answer: (D) lcm(a, b) = 2640

11. If lcm(a, b) = ab, then gcf(a, b)=

(A) a
(B) b
(C) ab
(D) 1
(E) None of these

Answer: (D) 1

12. If gcf(a, b) = a, then lcm(a, b)=

(A) a
(B) b
(C) ab
(D) 1
(E) None of these

Answer: (B) b

13. If gcf(a, b) = 1, then gcf(ac, bc) =

(A) a
(B) b
(C) c
(D) abc
(E) None of these

Answer: (C) c

14. If gcf(a, b) = 1, then lcm(ac, bc)=

(A) a
(B) b
(C) c
(D) abc
(E) None of these

Answer: (D) abc

15. The prime factorization of 280 is



(E) None of these

Answer:

16. Let where gcf(210, b) = 1. Which of the following numbers is not
a factor of a?

(A) 72
(B) 120
(C) 225
(D) 2520
(E) They are all factors of a.

Answer: (C)

17. Let where gcf(2310, b) = 1. Which one of the following numbers
is not a factor of a?

(A) 30
(B) 180
(C) 2310
(D) 3465
(E) They are all factors of a.

Answer: (E)

18. Let and . Then gcf(a, b)=



(E) None of these

Answer:

19. Let   and Then lcm(a, b) =



(E) None of these

Answer:

20. If , which of the following numbers could not be lcm(a, b)?



(E) They all could be lcm(a, b).

Answer: (A) 2 · 3 · 5 · 7

21. If , which of the following numbers could be gcf(a, b)?



(E) None of these

Answer: (A) 2 · 3 · 5 · 7

22. The number of positive factors of 120 is?

(A) 2
(B) 4
(C) 8
(D) 16
(E) None of these

Answer: (D) 16

23. The number of positive factors of 2940 is?

(A) 16
(B) 24
(C) 36
(D) 54
(E) None of these

Answer: (C) 36

24. If , then the number of positive factors of a is

(A) 360
(B) 480
(C) 600
(D) 840
(E) None of these

Answer: (A) 360

25. If , then the number of positive factors of a is

(A) 360
(B) 480
(C) 600
(D) 840
(E) None of these

Answer: (D) 840