UNIT I
GROUPS AND RINGS
1. All groups satisfy the properties from above A&B
1) G-i to G-v
2) G-i to G-iv
3) G-i to R-v
4) R-i to R-v
Ans: 2
2. An Abelian Group satisfies the properties from above A&B
1) G-i to G-v
2) G-i to R-iv
3) G-i to R-v
4) R-i to R-v
Ans: 1
3. A Ring satisfies the properties from above A&B
1) R-i to R-v
2) G-i to G-iv
3) G-i to R-v
4) G-i to R-iii
Ans: 4
4. A Ring is said to be commutative if it also satisfies the property from above A&B
1) R-vi
2) R-v
3) R-vii
4) R-iv
Ans: 4
5. An ‘Integral Domain’ satisfies the properties from above A&B
1) G-i to G-iii
2) G-i to R-v
3) G-i to R-vi
4) G-i to R-iii
Ans: 3
6.A Field satisfies all the properties from above A&B
1) G-i to G-iii
2) G-i to R-v
3) G-i to R-vii
4) G-i to R-iii
Ans: 3
7. What is the Associative property of a group.
Ans: 1
8. Homomorphism of groups (G,*) onto (G1,o) is defined as
Ans; 3
9. What is the Commutative property of a group.
Ans:3
10.Isomorphism of the two groups defined as f : G → G’ if
1. f is homomorphism and f is one-to-one
2. f is homomorphism and f is onto
3. f is homomorphism and f is epimorphism
4. f is homomorphism and f is one-to-one and onto
Ans: 4
11. Subgroup of Cyclic group is
1. Cyclic 2. Monoid 3. Field 4. Not cyclic
Ans:1
12. Any cyclic group is
1. Not a group 2. Semi group 3. Monoid 4. Abelian
Ans: 4
13. Left coset of the group is
Ans: 3
14. What is the Distributive property of a group.
Ans:3
15.For every group G, Identity element of G will be
1) More numbers 2) two 3) unique 4) its inverse
Ans:3
16) If and then AB in is
Ans:4
17) Let G be the group and let H be the sub group of group G, then
O(G) divides O(H)
O(H) divides O(G)
O(G) does not divides O(H)
O(H) does not divides O(G)
Ans:2
18) Given that is a cyclic group, then its generator is
2 2) 1 3) 4 4) 0
Ans:1
19) If is the field then if and only if n is
1) Composite number 2) prime number 3) non negative integer 4) complex number
Ans: 2
20) An finite ……….. is a field.
1) Group 2 ) cyclic group 3) ring 4) Integral domain
Ans: 3
21) Find the inverse of [25] in
1) [49] 2) [52] 3) [59] 4) )[25]
Ans:1
22) Find all x in such that
1) x=3 2) x=0 3) x=(2,3) 4) x= (1,4)
Ans:4
23) The……………..is always commutative.
1) Ring 2) Boolean Ring 3) Group 4) Sub group
Ans:2
24) The kernel of group homomorphism is
1) subgroup of the group
2) semi group
3) Normal subgroup of the group
4)Cyclic goup
Ans: 3
25) Every subgroup of abelian group is
1) Normal 2)Abelian 3)Cyclic 4)Monoid
Ans:1
26) If is group with usual multiplication then O(i) is
(Order of i in group G is)
1) 2 2)3 3)4 4)1
Ans:3
27. For every group G if the identity element of G is ….
a. Unique
b. Identity
c. Inverse
d. Element
Ans ;a
28. Every subgroup of a cyclic group is …..
a. subgroub
b. cyclic
c. group
d.not cyclic
Ans; b
29. Let f:G→H be a group homomorphism onto H. If G is abelian and H is …..
a.Group
b.abelian
c. subgroub
d. Permutation
Ans : b
30. Any cyclic group is ….
a. Group
b. abelian
c. subgroub
d. Permutation
Ans: b
31. An abelian group is …..
a. Group
b. Abelian
c. cyclic
d. not cyclic
Ans: d
32. Find the inverse of [25] in Z72
a. [49]
b.[72]
c.[25]
d.[85]
Ans: a
33. To find [100]-1 in the Ring Z1009
a. [222]
b. [333]
c. [111]
d. [444]
Ans:c
34. solve for x the linear congruence 3x≡7(mod 31)
a. x≡23(mod 31)
b.x≡57(mod 31)
c.x≡72(mod 31)
d.x≡113(mod 31)
Ans:a
35. solve for x the linear congruence 5x≡8(mod 37)
a. x≡10(mod 37)
b.x≡9(mod 37)
c.x≡11(mod 37)
d.x≡7(mod 37)
Ans:b
36. solve for x the linear congruence 6x≡97(mod 125)
a. x≡47(mod 125)
b.x≡27(mod 125)
c.x≡57(mod 125)
d.x≡37(mod 125)
Ans:d
37. In the group S5. Let ∝=1 2 3 2 3 1 4 5 4 5 and β=1 2 3 2 1 5 4 5 3 4 . Determine
a. 1 2 3 1 3 1 4 5 2 5
b. 1 2 3 1 5 2 4 5 4 3
c. 1 2 3 1 5 2 4 5 3 4
d. 1 2 3 1 5 3 4 5 4 2
Ans:c
38. In the group S5. Let ∝=1 2 3 2 3 1 4 5 4 5 and β=1 2 3 2 1 5 4 5 3 4 . Determine
a.1 2 3 1 2 3 4 5 4 5
b. 1 2 3 1 2 5 4 5 4 3
c. 1 2 3 1 3 2 4 5 5 4
d. 1 2 3 1 5 2 4 5 4 3
Ans:a
39. In the group S5. Let ∝=1 2 3 2 3 1 4 5 4 5 and β=1 2 3 2 1 5 4 5 3 4 . Determine
a.1 2 3 1 2 4 4 5 4 3
b. 1 2 3 2 3 1 4 5 4 5
c. 1 2 3 1 3 2 4 5 5 4
d. 1 2 3 2 1 3 4 5 4 5
Ans:d
40. In the group S5. Let ∝=1 2 3 2 3 1 4 5 4 5 and β=1 2 3 2 1 5 4 5 3 4 . Determine
a. 1 2 3 2 3 4 4 5 5 1
b. 1 2 3 2 1 3 4 5 4 5
c. 1 2 3 2 4 1 4 5 3 5
d. 1 2 3 2 1 4 4 5 5 3
Ans:b
41. . In the group S5. Let ∝=1 2 3 2 3 1 4 5 4 5 and β=1 2 3 2 1 5 4 5 3 4 . Determine ?
a. 1 2 3 2 1 4 4 5 5 3
b. 1 2 3 2 1 3 4 5 4 5
c. 1 2 3 2 4 1 4 5 3 5
d. 1 2 3 2 1 3 4 5 4 5
Ans:a
42. Every field is an …..
a. Integers
b. domain
c. codomain
d. integral domain
Ans:d
43. Any integral domain is a …..
a. Integral domain
b. domain
c. ring
d. field
Ans:c
44. Boolean ring is always ….
a. distributive law
b. Associative law
c. commutative law
d. inverse law
Ans:a
45. For the group Sn of all Permutations of n distinct symbol what is the number of elements in Sn ?
a. n
b.n-1
c. 2n
d. n!
Ans:d
46. Inverse of –I in the multiplicative group, {1 , -1, i, -i}
a.1
b.-1
c. i
d.-i
Ans:c
47. In Zn , [a] is a unit iff gcd(a,n)=
a.1
b. 0
c. a
d. n
Ans: a
48.All integers n exactly one of n, 2n-1 and 2n+1 is divisible by
a. 2
b.3
c. 4
d.5
Ans:b
49. What is the last digit in 355?
a. 7
b. 5
c. 0
d.9
50. How many rings are there in the ring Z8?
a. 4
b.6
c. 3
d. 2
Ans:a
51. How many units are there in the ring Z2 X Z2 X Z2?
a. 1
b.2
c. 6
d. 8
Ans:a
Unit II
Finite Fields and polynomials
Degree of non zero constant polynomial is
0 b) 1 c) constant d) not defined
Ans:a
If R is the Ring with identity 1, then R[x] is a ring with identity
0 b) 1 c) e d) R
Ans: b
If R is an integral domain , then deg(f(x).g(x))=
deg ( f(x)+g(x)) b) deg f(x)+deg g(x) c) deg f(x).deg g(x) d) deg f(x)-deg g(x)
Ans :b
An element a is root of f(x)€ R[x] if
f(x)=a b) f(1)=1 c) f(a)=0 d) f(a)=1
Ans:c
Find the roots of the polynomial x2-2€ Q[x]
No roots b) 2,-√2 c) 1,-1 d) 2,-2
Ans:a
Find the roots of the polynomial x2-2€ R[x]
No roots b) 2,-√2 c) 1,-1 d) 2,-2
Ans:b
7. What is the degree of the polynomial f(x)=6x3+5x2+3x-2 over Z6?
a) 3 b) 1 c) 2 d) 0
Ans:c
8. In Division algorithm for polynomials g(x)=q(x)f(x)+r(x), the value of r(x) will be
a) r(x)=0 or deg r(x) <deg f(x) b) r(x)=0 or deg r(x) >deg f(x)
c) r(x)=0 or deg r(x) <deg g(x) d) r(x)=0 or deg r(x) >deg g(x)
Ans:a
9.Let F be a Field ,a€F and f(x) € F(x) then f(a) is remainder when
a) (x-a) is factor of f(x) b) f(x) is factor of (x-a)
c) f(x) is divided by (x-a) d) ) (x-a) is divided by f(x)
Ans:c
10. Let F be a Field ,a€F and f(x) € F(x) then a is root of f(x) iff
a) (x-a) is factor of f(x) b) f(x) is factor of (x-a)
c) f(x) is divided by (x-a) d) ) (x-a) is divided by f(x)
Ans:a
11. What is the remainder when f(x)=x5+ 2x3+x2+2x+3 € Z5(x) is divided by (x-1)
a) 6 b) 12 c) 4 d) 16
Ans:c
12. What is the remainder when f(x)=2x3+x2+2x+23€ Z5(x) is divided by (x-2)
a) 0 b) 2 c) 3 d) 4
Ans:b
13. If f(x) €F(x) is of degree n≥1, then
a) f(x) has at most n roots in F b) f(x) has at most n-1 roots in F
c) f(x) has at most n+1 roots in F d) f(x) has at most n-2 roots in F
Ans:a
14.Let f(x)=g(x)h(x) be a reducible polynomial then deg(g(x)) and deg(h(x)) are
a) deg(g(x))≤1 and deg(h(x)) ≤1 b) deg(g(x))≥1 and deg(h(x)) ≥1
c) deg(g(x))=0 and deg(h(x)) =0 d) deg(g(x)) ≥1 and deg(h(x)) ≤1
Ans:b
15. Find the factor in f(x) =x3+x2+x+1€ Z2(x)
a) (x-1) b) (x) c) (x-2) d) (x+1)
Ans:a
16. Let F be a field and f(x) €F(x).If f(x) is of degree 1, then
a) f(x) is has root b) f(x) is integral doamin c)f(x) is reducible d) f(x) is irreducible
Ans: d
17.Let F be a field and f(x),g(x) €F(x). If d(x) is the g.c.d of f(x) and g(x) then d(x)
a)d(x) divides f(x) alone b) d(x) divides g(x) alone
c) d(x) divides both f(x) and g(x) d) d(x) does not divides both f(x) and g(x)
Ans:c
18. In Division algorithm of two polynomials f(x) and g(x), the last non zero
remainder is called
a) l.c.m of (f(x),g(x)) b) g.c.d(f(x),g(x))
c) zero polynomial d) Multiplication of f(x) and g(x)
Ans:b
19. The number of elements of a finite field is……., where p is prime number and n is a
Positive integer
a) pn b) pn-1 c) pn-2 d) p1-n
Ans:a
20. An ideal is always……..
a) Ring b) Subring c)Integral domain d)Field
Ans: b
21.How many polynomials are there of degree 2 in Z11(x)?
a).1012 b)1102 c)1210 d)2101
Ans: c
22.If (2x+1) is unit in Z4(x) then find its inverse in Z4(x).
a)2x+2 b)2x+1 c)2x+3 d)4x+1
Ans:b
23.In monic polynomial the leading coefficient is……
a) 1 b)2 c)-1 d)-2
Ans: a
24.The polynomial f(x)=2x2+4 is irreducible over
a) R b) Q c) Z d) C
Ans:d
25. The characteristic of a Field (F,+, .) is either 0 or
a) 1 b)constant c)prime d) composite
Ans:c
26. If R is a ring then (R(x),+,.) is
ring b) not a ring c) Field d) integral domain
Ans:a
If R[x] is integral domain then R is
ring b) not a ring c) Field d) integral domain
Ans: d
If f(x) belongs to F(x) and sF, then then remainder in the division of f(x) by x-s i s
f(a) b) f(x) c) f(s) d) f(x-s)
Ans :c
Find the roots of the polynomial x2-2€ Q[x]
No roots b) 2,-√2 c) 1,-1 d) 2,-2
Ans:a
If f(x) F(x) and a F then x-a is the factor of f(x) iff the root of f(x) is
1 b) 2 c) a d) x-a
Ans:c
31. if f(x) F(x) has degree n 1 then f(x) has atmost _____ roots
a) n b) n-1 c) n+1 d) 0
Ans:a
32. In Division algorithm for polynomials g(x)=q(x)f(x)+r(x), the value of r(x) will be
a) r(x)=0 or deg r(x) <deg f(x) b) r(x)=0 or deg r(x) >deg f(x)
c) r(x)=0 or deg r(x) <deg g(x) d) r(x)=0 or deg r(x) >deg g(x)
Ans:a
33.How many polynomials are there degree n in Z11(x)?
a) 10. 11n b) 11. 10n c) 10. 11n-1 d) ) 11. 10n-1
Ans:a
34. If is divided by then Q(x) is ___
a) x2+5 b) x+5 c) 25x3-30x d) 25x3 -9x2 -30x-3
Ans:b
35. If is divided by then r(x) is ___
a) x2+5 b) x+5 c) 25x3-30x d) 25x3 -9x2 -30x-3
Ans:d
36. What is the remainder when f(x)=2x3+x2+2x+23€ Z5(x) is divided by (x-2)
a) 0 b) 2 c) 3 d) 4
Ans:b
37. One of the roots of x4-16 is ____
a) 2i b) 4i c) -4i d) i
Ans:a
38. What is the remainder when f(x)=x100+x90+x80+x50+1 divided by (x-1)
a) 0 b) 2 c) 1 d) 4
Ans:c
39. How many units are there in the ring Zp(x), p a prime?
a) p-1 b) p+1 c) p d) p-n+1
Ans:a
40. Find the factor in f(x) =x3+x2+x+1€ Z2(x)
a) (x-1) b) (x) c) (x-2) d) (x+1)
Ans:a
41. If R is an intergral domain , if f(x) is a unit in R(x) then f(x) is a constant and _____in R
a) ideal b) unit c)Field d) zero polynomial
Ans: b
42.Every non-zero polynomial of degree 1 is
a)reducible b) Irreducible c) unit d) Ring
Ans:a
43. If f(x), g(x) F(x) and their GCD is 1 then f(x) and g(x) are called
a) Prime b) Not a prime
c) zero polynomial d) Relatively prime
Ans:d
44. The number of elements of a finite field is……., where p is prime number and n is a
Positive integer
a) pn b) pn-1 c) pn-2 d) p1-n
Ans:a
45. Let ( F, +, .) be a field, If char F > 0, then Char(F) must be _____
a) Ring b) Subring c)Integral domain d)Prime
Ans: d
46.If s(x) is irreducible in F(x) then F(x)/s(x) is a _____
a).ring b)Field c)Integral domain d)Principal ideal
Ans: b
47.The Characteristic of Z11 is
a)22 b)11 c)33 d)10
Ans:b
48.In monic polynomial the leading coefficient is……
a) 1 b)2 c)-1 d)-2
Ans: a
49.If s(x) = x4+x3+1, the order of Z(x)/s(x) is
a) 4 b) 8 c) 16 d) 2
Ans:c
50. The characteristic of Q(x) is
a) 1 b)0 c)prime d) composite
Ans:b
Unit-3
Divisibility Theory and Canonical Decompositions
1. State the division algorithm for two integers a and b, where b>0.
(a) a=qb+r, 0≤r<b.
a=qb-r, 0≤r<b.
a=qb-r, 0≥r>b.
a=qb+r, 0≥r>b
Ans:a
2. Find the quotient and remainder in the division algorithm when -23 is
divided by 5.
(a) -23=5(5)+2, 0≤2<5
(b) -23=-5(5)-2, 0≤2<5
(c) -23=-5(5)+2, 0≤2<5
(d) 23=-5(5)+2, 0≤2<5
Ans:c
3. If a divides b and a divides c then a divides -2c+3b , So its true and false.
(a) True (b) False
Ans : a
4. Find the number of positive integers 2076 that are divisible by 19.
(a) 108 (b) 106 (c) 107 (d) 109
Ans : d
5. If the square of an integer is even , then the integer is
(a) Odd (b) Even (c) Prime (d) neither even nor odd
Ans : b
6. Find the number of positive integers 2076 that are divisible by neither 4 nor 5.
. (a) 1244 (b) 1243 (c) 1245 (d) 1242
Ans : c
7. Find the gcd of 168 and 180 using the canonical decomposition.
(a) 11 (b) 13 (c) 10 (d) 12
Ans : d
8. Find the canonical decomposition of 29 - 1
(a) 511 (b) 510 (c) 506 (d) 509
Ans : a
9. If a divides b and c divides d then
(a) ab divides cd (b) ac divides bd (c) ad divides bc (d) a didvides d
Ans: b
10. Express (10110)2 in base 10
(a) 20 (b) 21 (c)22 (d) 23
Ans : c
11. Express (1776)8 as a decimal number
(a) 1020 (b) 1019 (c)1021 (d) 1022
Ans : d
12. Write 111010 2 as an octal integer
(a) 728 (b) 722 (c) 7210 (d) 726
Ans : a
13. Write 1101012 as a hexadecimal number
(a) 7516 (b) 7512 (c) 7515 (d) 7514
Ans : a
14. Find the value of the base b if 54b = 64
(a) 11 (b)12 (c) 10 (d) 13
Ans : b
15. If (a , b) = d , then
(a) (a/b , b/d ) = 1 (b) (a/d , b/d ) = 2 (c) (a/d , b/d ) = 1 (d) (a/b , b/d ) = 1
Ans : c
16. If n is a positive integer , then gcd ( n , n+2) is
(a) 1 or 3 (b) 2 or 3 (c) 3 or 4 (d) 1 or 2
Ans : d
17. If a , b , c are positive integer, then gcd (ca , cb) = ?
(a) c gcd(a,b) (b) b gcd(a,c) (c) a gcd(b,c) (d) gcd (a , b)
Ans : a
18. If (a,b) =1 then ( a+b , a-b ) = ?
(a) 1 or 3 (b) 2 or 3 (c) 3 or 4 (d) 1 or 2
Ans: d
19. If (a,b)=3 and ab = 693 find[ a, b ]
(a) 231 (b) 230 (c) 229 (d) 228
Ans : a
20. If a and b are positive integer with a = 231, (a, b) = 7 and [a,b] =600060, find b.
(a) 1819 (b) 1820 (c) 1821 (d) 1822
Ans : b
21. If (a,b) = [a,b] what can you say about the relation between a and b.
(a) a> b (b) a< b (c) a = b (d) ab
Ans : c
22. Find the positive integer a if [a , a+1] = 132
(a) 10 (b) 12 (c) 13 (d) 11
Ans : d
23. If ab =156 , a and b are relatively prime find [a,b]
(a) 156 (b) 155 (c) 154 (d) 153
Ans : a
24. The Lcm of two consecutive positive integers is 812 find the number
(a) 27 (b) 28 (c) 26 (d) 25
Ans : b
25. If a and b are positive integers with a.b = 24 . 34 . 5 . 7 . 113 . 13 and
[a,b]= 22 . 33. 52 . 7 . 112 . 13 then what is (a,b)=
(a) 329 (b) 328 (c) 330 (d) 331
Ans : c
26. If f(n) denotes the number of positive factors of a positive integer n, then f(12)=
(a) 6
4
8
11
Ans:a
27. Let a,b Z such that a/b and b/a then
(a) a b
(b) a=b
(c) a/a
(d) b/b
Ans:b
28 The no of positive integers 2076 and divisible by neither 4 nor 5 is
(a) 831 (b) 1234 (c) 1245 (d) 830
Ans : c
29. The sum of the product of any two even integers are.
(a) even (b) odd (c) neither odd nor even (d) prime
Ans : a
30. If the square of an integer is even , then the integer is
(a) Odd (b) Even (c) Prime (d) neither even nor odd
Ans : b
31. 2n3+3n2+n is divisible by.
. (a) 2 (b) 3 (c) 6 (d) 12
Ans : c
32. Expansion of 10110two in base ten is
(a) 25 (b) 24 (c) 23 (d) 22
Ans : d
33. Expansion of 1776eight in base ten is
(a) 1022 (b) 1023 (c) 1024 (d) 1025
Ans : a
34. Find b, if 1001b = 9
(a) 3 (b) 2 (c) 5(d) 4
Ans: b
35. Find a if 144a= 49
(a) 6 (b) 4 (c)5 (d) 9
Ans : c
36. The Number of ones in the binary representation of 24-1 is
(a) 6 (b) 5 (c)4 (d) 3
Ans : c
37. Every composite number n has a prime factor
(a) ≤√n (b) ≥√n (c) <√n (d) >√n
Ans : a
38. The composite six consecutive integers are
(a) 5043,5044,5045,5046,5047,5048
(b) 5042,5043,5044,5045,5046,5047
(c) 5052,5053,5054,5055,5056,5057
(d) 5041,5042, 5043,5044,5045,5046
Ans : b
39.If p and q are successive odd primes and p+q = 2r then r is _____
(a) prime (b)composite (c) twinprime (d) relatively prime
Ans : b
40. Any two consecutive Fibonacci numbers are
(a) prime (b)composite (c) twinprime (d) relatively prime
Ans : d
41. If n is a positive integer , then gcd ( n , n+2) is
(a) 1 or 3 (b) 2 or 3 (c) 3 or 4 (d) 1 or 2
Ans : d
42. If a , b , c are positive integer, then gcd (ca , cb) = ?
(a) c gcd(a,b) (b) b gcd(a,c) (c) a gcd(b,c) (d) gcd (a , b)
Ans : a
43. If b=a2 then (a,b) is
(a) b (b) a (c) a2 (d) a3
Ans: b
44. If d = (a,b) and d1 is any common divisor of a and b then
(a) d/d1 (b) d1/d (c) d does not divides d1 (d) d1 does not divides d
Ans : b
45. (18,30,60,75,132)=.
(a) 3 (b) 6 (c) 9 (d)12
Ans : a
46. Every integer n≥2 either is a prime or can be expressed as a product of_____
(a) primes (b) composites (c) integers (d) real numbers
Ans : a
47. The positive factors of 48 is
(a) 10 (b) 12 (c) 8 (d)6
Ans : a
48. There are infinitely many primes of the form_____
(a) 4n+3 (b) 4n-3 (c) 4n+2 (d) 4n-2
Ans : a
49. [1050,2574]=
(a) 350450 (b) 450450 (c) 440450 (d) 350350
Ans : b
50. If a and b are positive integers with a.b =
(a) a[a,b] (b) a(a,b) (c) (a,b)[a,b] (d) ab[a,b]
Ans : c
ALGEBRA AND NUMBER THEORY
Unit IV- Diophantine equations and Congruences
The Linear Diophantine equation ax+by=c is solvable where d=(a,b) if
a divides c b) c divides d c) b divides c d) d divides c
Ans: d
then
b) c) d) -
Ans: b
One of the solution of Hundred Fowl’s Puzzle is
b)
c) d)
Ans: a
If iff
b) c) d)
Ans: a
(175 , 192) is equal to
16 b) 1 c) 8 d) 12
Ans: b
[252, 360] is equal to
100 b) 3310 c) 2310 d) 2520
Ans :d
If the LDE is solvable then the solution is
a) b)
c) d)
Ans:a
8.In Euclidean algorithm of a & b, the last nonzero remainder is
a) gcd of a & b b) lcm of a & b c) prime number d) linear combination of a & b
Ans: a
9. One of the solution of Mahavira’s Puzzzle is
a) b) c) d)
ans: b
10. Which one is the solution of linear combination of 18 and 28
a) b) c) d)
Ans: c
11. Find the remainder a is divisible by 7.
a)3 b) 6 c) 8 d) 12
Ans:a
12. The LDE 5x+7y = 13 can be written as
a) b) c) d)
Ans: b
13. Find the remainder a is divisible by 15.
a) 3 b) 4 c)5 d)6
Ans : b
14. The solution of Chinese Remainder Theorem is
a) b)
c) d)
Ans:c
15. -23 Div 25 is equal to
a) -5 b) 5 c) -1 d) 1
Ans: c
16. If the LDE is 12x+13y = 14. Then d is equal to
a) 6 b) 12 c) 48 d) 1
Ans:d
17. Ififf a and b leaves the _______
a) same quotient b) same remainder c) Different quotients d) Different remainders
Ans: b
18. If and
a) b) c) d)
ans:a
19. When is divided by 11 then the remainder is
a) 32 b) 9 c) 3 d) 0
Ans:d
20. Find the remainder a is divisible by 12.
a) 0 b) 1 c) Infinite d) No solution
Ans:b
21. then
a) m divides a-b b) a+b divides m c) a-b divides m d) m divides a+b
Ans: a
22. The Linear congruence is solvable iff d divides b where d is equal to
a) (a,b) b) (b,m) c) (a,m) d) (ax,bm)
Ans; c
23. has _____ incongruent solutions
a) 2 b) 6 c) 3 d) 9
Ans:b
24. The linear systems has unique solution if
then the value of is
a) b) c) d)
Ans: d
25. An integer is divisible by 9 iff the sum of the digits is divisible by
a) 3 b) 9 c) 6 d) 3 and 9
is divisible by
12 b) 14 c) 16 d) 22
Ans:a
is congruent to
-10 b) 10 c) 4 d) -4
Ans;c
One of the solution of Hundred Fowl’s Puzzle is
b) c) d)
Ans:a
If iff
b) c) d)
Ans:c
(120 , 28) is equal to
16 b) 4 c) 8 d) 12
Ans:a
[110, 210] is equal to
10 b) 3310 c) 2310 d) 2210
Ans:c
If the LDE is solvable then the solution is
a) b)
c) d)
Ans:a
33. The LDE is solvable iff
a) c divides d b) d divides c c) c does not divides d d) d does not divides c
Ans:b
34. One of the solution of Mahavira’s Puzzzle is
a) b) c) d)
Ans:d
35. Which one is the solution of linear combination of 28 and 12
a) b) c) d)
Ans:c
36. The congruence has ……. Solutions
a) 4 b) 6 c) 8 d) 12
Ans:b
37. The LDE 2x+3y = 5 can be written as
a) b) c) d)
Ans:b
38. The congruence’s where the module are Pairwise relatively prime has______ solutions
a) Indefinite b) Finite c) Unique d) No solutions
Ans:c
39. The solution of Chinese Remainder Theorem is
a) b)
c) d)
Ans:C
40. -72 Div 17 is equal to
a) -55 b) 55 c) -4 d) 4
Ans:c
41. If the LDE is 12x+16y = 20. Then d is equal to
a) 6 b) 12 c) 48 d) 4
Ans:d
42. Ififf a and b leaves the _______
a) same quotient b) same remainder c) Different quotients d) Different remainders
Ans:b
43. In the Euclidean Algorithm Method, _______ is the GCD
a) Last remainder b) Last non-zero remainder c) First remainder d) First non-zero remainder
Ans:b
44. When is divided by 15 then the remainder is
a) 32 b) 9 c) 3 d) 0
Ans:c
45. If a is divisible by b then the remainder is
a) 0 b) 1 c) Infinite d) No solution
Ans:a
46. a divides by b is
a) a divided by b b) b divided by a c) 0 d) Indefinite
Ans:b
47. The Linear congruence is solvable iff d divides b where d is equal to
a) (a,b) b) (b,m) c) (a,m) d) (ax,bm)
Ans:c
48. has _____ incongruent solutions
a) 2 b) 6 c) 3 d) 9
Ans:b
49. If , then
a) b) c) d)
Ans:d
50. If is divided by 7 then the remainder is
a) 1 b) 2 c) 6 d) 5
Ans:bALGEBRA AND NUMBER THEORY
Unit V- Classical Theorems and Multiplicative Functions
If a is self invertible if and only if
b) c) d)
Ans:d
In Wilson’s theorem ,If p=2 then,
a) b)
c) d)
Ans:a
A positive integer p is prime iff ∅p=
p-2 b) p-1 c) 0 d) 1
Ans:b
Let p be a prime and a is any integer such that p does not divides a
is an inverse of modulo p b) is an inverse of modulo p c) is an inverse of modulo p d) is an inverse of modulo p
Ans: b
If the one of the solution is
b) c) d)
Ans:a
let m be a positive integer, then defined as
a) number of positive integers and are relatively prime to m
b) number of positive integers and are relatively prime to m
c) sum of positive integers and are relatively prime to m
d) sum of positive integers and are relatively prime to m
Ans:a
7. Let p be a prime number and let a be any integer. Then
a) p (mod a) b) a (mod p) c) -1(mod p) d) 1(mod p)
Ans:d
8. Let m be the positive integer and a is any integer with (a,m)=1, then
a) b) c) d)
Ans:c
9. Find the remainder when is divided by 18
a) 13 b) 14 c) 15 d) 18
Ans:a
10.Find
a) 6 b) 12 c) 4 d) 8
Ans:c
11. Find
a) b) c) d)
Ans:d
12. Find
a) b) c) d)
Ans:a
13. Let p be a prime, then
a) b) c) d)
Ans:b
14. Find
a) 36 b) 26 c) 16 d) 32
Ans:d
15. Compute , for n=18
a) 48 b) 20 c) 10 d) 18
Ans:d
16.If p and q are distinct primes,
a) 1(mod pq) b) 0(mod pq) c) -1(mod pq) d) pq(mod pq)
Ans:a
17. If p and q are distinct primes
a) pq(mod pq) b) p+q(mod pq) c) p-q(mod pq) d) p/q(mod pq)
Ans:b
18.Find the twin Primes p and q if
a) p=11,q=13 b) p=11,q=17 c) p=17,q=13 d) p=19,q=23
Ans:a
19. If n is odd then
a) b) c) d)
Ans: c
20. If n is even then
a) b) c) d)
Ans:d
21.Find the one digit in the base seven expansion of
a) 4 b) 2 c) 1 d) 3
Ans:d
22. Find
a) 9 b) 18 c) 12 d) 3
Ans:c
23.Find
a) 0 b) 13 c) 1 d) 2
Ans:d
24.Let p be any prime and e be any positive integer.Then
a) e+1 b) e-1 c) p-1 d)p+1
Ans:a
25. .Let p be any prime and e be any positive integer.Then
a) b) c) d)
Ans:b.
If p is prime, then (p-1)! =
-1(mod p) b) 1(mod p) c) 0(mod p) d) 1!(mod p)
If p be any prime and n any positive integer, then np!n!pn
0(mod p) b) 1(mod p) c) -1(mod p) d) p!(mod p)
A positive integer p is prime iff ∅p=
p-2 b) p-1 c) 0 d) 1
let n be a positive integer, then σ(n) is defined as
sum of positive factors of n b) number of positive factors of n c) n! d) (n-1)!
If f is a multiplicative function, then F(n) =
d/nf(d) b) d/nf(n) c) n/df(d) d) n/df(n)
let n be a positive integer, then (n) is defined as
a)sum of positive factors of n b) number of positive factors of n c) n! d) (n-1)!
32. let n≥2 be a positive integer, then n is prime iff
a) (n+1)! 1 (mod n) b) (n-1)! 1 (mod n) c) (n+1)! -1 (mod n) d) (n-1)! -1 (mod n)
33. Let p be a prime, then (p-1)(p-2)…(p-k) congruent to
a) -1kk!(mod k) b) -1kk!(mod p) c) -1kp!(mod k) d) -1pk!(mod k)
34.Let p be a prime number and let a be any integer. Then ap
a) p (mod a) b) a (mod p) c) -1(mod p) d) 1(mod p)
35. If p and q are distinct primes, then pq-1+qp-1
a) -1(mod pq) b) 1 (mod pq) c) p (mod pq) d) q(mod pq)
36. 12=
a) 6 b) 12 c) 10 d) 28
37. 36=
a) 9 b) 6 c) 12 d) 18
38. Let a be a solution of the congruence , then the another solution is
a) m+a b) m-a c) m+1 d) m-1
39. Let p be an odd prime, then
a) 0(mod p) b) 1(mod p) c) -1(mod p) d) p-1(mod p)
40. =
a) 8 b) 4 c) 2 d) 0
41. Compute , for n=10
a) 48 b) 20 c) 10 d) 8
42.If
a) 1(mod ab) b) 0(mod ab) c) a(mod ab) d) b(mod ab)
43. If , then =
a) n/2 b) k/2 c) n-1 d) k-1
44.Let fn denote a Fermat Prime, then =
a) fn-1 b) fn+1 c) fn-2 d)fn+2
45. Let p be a prime and a any integer such that p does not divides a. then
a) 1 (mod p) b) 0 (mod p) c) p-1(mod p) d) p+1(mod p)
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