# Solve Number Theory Mcqs | Number Theory Mcqs with solution

## Solve Number Theory Mcqs | Number Theory Mcqs with solution

1. For any positive integers a and b, there exists a positive integer n such that na > b is called:

A. **Archimedean Property**

B. Division Algorithm

C. Density Theorem

D. Fundamental Theorem of Arithmetic

2. Let S ⊆ N having the properties:

(i) 1 ∈ S and

(ii) Whenever k ∈ S, then k + 1 ∈ S, then

A. **S = N**

B. S ⊆ N

C. S ⊇ N

D. S 6= N

3. 2[1 + 2 + 3 + …. + n]=

A. n(n+1)/2

B. n(nn 1)/2

C. **n(n + 1)**

D. n(n – 1)

4. Given integers a and b with b 6= 0, there exist unique integers q and r satisfying

A**. a = bq + r, 0 ≤ r < |b|**

B. a = bq + r, 0 ≤ q < |b|

C. a = bq + r, 0 ≤ r < |a|

D. a = bq + r, 0 ≤ q < |a|

5. Which of the following is false?

A. a|a

B. If a|b and b|c, then a|c

C. **If a|b and b|a, then a = b**

D. If a|b then a|bc

6. If a|b and a|c, then for any x, y ∈ Z, we have

A. a|(bx + cy)

B. a|(bx x cy)

C. a|bc

D. **All of these**

7. If a|(b + c) and a|b, then

A. **a|c**

B. a – c

C. a|(b – c)

D. a – (b – c)

8. If a = 73 and b = 8, then

A. q = 9, r = =1

B**. q = 9, r = 1**

C. q = -9, r = 1

D. q = -9, r = -1

9. If a = 23 and b = 7, then

A. q = 4, r = 5

B. q = -4, r = 5

C. q = 4, r = -5

D. q = -4, r = -5

10. We read a|b as

A. a divides b

B. b is divisible by a

C. b is multiple of a

D. **All of these**

11. Let a, b ∈ Z with a ≠ 0. Then a|b if for some c ∈ Z,

A. a = bc

B. **b = ac**

C. c = a + b

D. c = ab

12. Any integer can be expressed in the form

A. 2n or 2n + 1

B. 3n, 3n + 1 or 3n + 2

C. 4n, 4n + 1, 4n + 2 or 4n + 3

D. **All of these**

13. For any n ∈ Z, 2.7^n + 3.5^n – 5 is divisible by

A. **24**

B. 23

C. 9

D. 13

14. The product of any three consecutive integers is divisible by

A. 4

B. 5

C**. 6**

D. 7

15. Let a, b be nonzero integers. Then a positive integer d is called … of a and b if

(i) d|a and d|b

(ii) If c|a and c|b, then c ≤ d.

A. G. C. D

B. L. C. M

C. H. C. F

D. **Both A and C**

16. Let a, b be nonzero integers and (a, b) = 1, then a, b are called

A. Prime to each other

B. Coprime

C. Relatively prime

D. **All of these**

17. The G.C.D of two non zero integers a and b:

A. Is always unique

B. Is not necessarily unique

C. Always exists

D. **Both A and C**

18. If a|b, then (a, b)=

A. **a**

B. b

C. |a|

D. |b|

19. (8, -40)=

A**. 8**

B. -8

C. 2

D. -2

20. If d = (a, b), then there exist x, y ∈ Z such that:

A. d = ax + by

B. d = ax x by

C. d = ay + bx

D. **All of these**

21. Let k ∈ Z and a, b ∈ Z − {0}

A. k(a, b)

B. **|k|(a, b)**

C. Both A and B

D. None of these

22. If d = (a, b), then

A**. ( a/d,b/d) = 1**

B. ( a/d,b/d) = d

C. ( a/b,b/a) = d

D. ( a/b,b/a) = 1

23. If a|bc and (a, b) = 1, then

A. **a|c**

B. b|c

C. a – c

D. a|(b + c)

24. Let a, b ∈ Z − {0}. Then a positive integer m is called … of a and b if

(i) a|m and b|m

(ii) If a|n and b|n then m ≤ n.

A. G. C. D

**B. L. C. M**

C. H. C. F

D. Both B and C

[We denote L. C. M of a and b as < a, b >, [a, b] or lcm(a, b).]25. For any non zero integers a, b we have

A. < a, b >= ab(a, b)

B. (a, b) = ab < a, b >

C. a(a, b) = b < a, b >

D. **< a, b > (a, b) = ab**