Number Theory (Final Exam Bank) - 175 Questions
Department of Cyber Security - Al-Mustaqbal University
Prepared by Student: Hussein Abdullah Rahman
Specialization: Cyber Security (First Stage)
University: Al-Mustaqbal University
Subject: Number Theory
Specialization: Cyber Security (First Stage)
University: Al-Mustaqbal University
Subject: Number Theory
Q1. Which branch of mathematics is often referred to as the Queen of Mathematics?
A. Calculus
B. Number Theory
C. Geometry
D. Algebra
E. Topology
Q2. Which ancient civilization used number systems for practical calculations?
A. Babylonians and Egyptians
B. Romans
C. Greeks only
D. Mayans
E. Incas
Q3. Who developed fundamental theorems on divisibility and prime numbers?
A. Fermat
B. Euler
C. Gauss
D. Euclid
E. Pythagoras
Q4. Who is known for the Little Theorem and his famous Last Theorem?
A. Euclid
B. Gauss
C. Pierre de Fermat
D. Euler
E. Turing
Q5. Who expanded number theory through the totient function?
A. Fermat
B. Leonhard Euler
C. Euclid
D. Gauss
E. Newton
Q6. Who introduced modular arithmetic and the theory of congruences?
A. Euclid
B. Euler
C. Fermat
D. Carl Friedrich Gauss
E. Pascal
Q7. RSA encryption and elliptic curve cryptography rely on the properties of:
A. Calculus integrals
B. Geometric shapes
C. Prime numbers and modular arithmetic
D. Differential equations
E. Complex analysis
Q8. Hash functions and random number generation are applications of Number Theory in:
A. Biology
B. Coding Theory
C. Computer Science
D. Chemistry
E. Statistics
Q9. Applications in data transmission and error correction belong to:
A. Number Theory history
B. Cryptography
C. Computer Science
D. Coding Theory
E. Algebra
Q10. The sum of consecutive odd numbers forms:
A. Prime numbers
B. Perfect squares
C. Even cubes
D. Negative numbers
E. Fractions
Q11. What is the sum of 1 + 3 + 5 + 7 + 9?
A. 16
B. 25
C. 36
D. 49
E. 64
Q12. What is the sum of 1 + 3 + 5 + 7 + 9 + 11 + 13?
A. 25
B. 36
C. 49
D. 64
E. 81
Q13. What is the product of 111 × 111?
A. 121
B. 12321
C. 1234321
D. 123454321
E. 111111
Q14. What is the product of 11111 × 11111?
A. 12321
B. 1234321
C. 123454321
D. 12345654321
E. 111111111
Q15. What is the value of 4! ?
A. 6
B. 24
C. 120
D. 720
E. 5040
Q16. What is the value of 5! ?
A. 24
B. 120
C. 720
D. 5040
E. 40320
Q17. What is the value of 6! ?
A. 120
B. 720
C. 5040
D. 40320
E. 362880
Q18. What is the value of 7! ?
A. 720
B. 5040
C. 40320
D. 362880
E. 3628800
Q19. What is the value of 8! ?
A. 5040
B. 40320
C. 362880
D. 3628800
E. 120
Q20. In Pascal's Triangle, the sequence 1, 4, 6, 4, 1 represents the coefficients for the power of:
A. 2
B. 3
C. 4
D. 5
E. 6
Q21. A well-defined collection of distinct objects enclosed in curly brackets is called a:
A. Group
B. Ring
C. Set
D. Field
E. Matrix
Q22. The set {1, 2, 3, 4, 5} is an example of a:
A. Finite Set
B. Infinite Set
C. Null Set
D. Universal Set
E. Empty Set
Q23. The set {1, 2, 3, ...} is an example of an:
A. Finite Set
B. Infinite Set
C. Null Set
D. Empty Set
E. Unit Set
Q24. What does the symbol ∅ represent in set notation?
A. Universal Set
B. Infinite Set
C. Empty Set
D. Subset
E. Superset
Q25. In set notation, the symbol ∈ denotes:
A. Subset of
B. Not an element of
C. Element of
D. Intersection
E. Union
Q26. What does the set N represent?
A. Integers
B. Natural numbers
C. Rational numbers
D. Real numbers
E. Complex numbers
Q27. What does the set Z represent?
A. Natural numbers
B. Real numbers
C. Integers
D. Rational numbers
E. Prime numbers
Q28. The set Z consists of:
A. Positive integers only
B. Negative integers only
C. Zero only
D. Positive, negative integers, & zero
E. Fractions
Q29. The property stating that for any a, b ∈ N, the sum a + b is also in N is called:
A. Associativity
B. Commutativity
C. Closure
D. Distributivity
E. Identity
Q30. The property (a + b) + c = a + (b + c) is known as:
A. Closure
B. Associativity
C. Commutativity
D. Cancellation
E. Identity
Q31. The property a + b = b + a is known as:
A. Associativity
B. Closure
C. Commutativity
D. Distributivity
E. Cancellation
Q32. If a + c = b + c, then a = b. This is known as the:
A. Associativity law
B. Distributivity law
C. Identity law
D. Cancellation Law
E. Closure law
Q33. What is the multiplicative identity in the set of natural numbers (N)?
A. 0
B. 1
C. -1
D. 2
E. None
Q34. The property a(b + c) = ab + ac represents:
A. Associativity
B. Commutativity
C. Distributivity
D. Closure
E. Cancellation
Q35. Is the operation of subtraction always closed in the set of natural numbers (N)?
A. Yes, always
B. No, not always
C. Only for primes
D. Only for evens
E. Only if a > b
Q36. Is the operation of division generally closed in the set of natural numbers (N)?
A. Yes, always
B. No
C. Only for positive outcomes
D. Only for primes
E. Yes, except zero
Q37. What is the additive identity in the set of integers (Z)?
A. 0
B. 1
C. -1
D. 2
E. None
Q38. In the set of integers (Z), every element 'a' has an additive inverse denoted as:
A. 1/a
B. |a|
C. -a
D. a^2
E. a!
Q39. Is the operation of subtraction always defined (closed) in the set of integers (Z)?
A. Yes, always defined
B. No
C. Only for positive values
D. Only for negative values
E. Only if a > b
Q40. What is the multiplicative identity in the set of integers (Z)?
A. 0
B. 1
C. -1
D. e
E. None
Q41. Is the operation of division a closed operation in the set of integers (Z)?
A. Yes
B. No
C. Only for positive integers
D. Only for negative integers
E. Always except zero
Q42. For any a, b ∈ Z, what is the result of a × 0?
A. a
B. 1
C. 0
D. -a
E. Undefined
Q43. For any a, b ∈ Z, what is the result of (-a)b ?
A. ab
B. -ab
C. a-b
D. 0
E. a+b
Q44. For any a, b ∈ Z, what is the result of (-a)(-b) ?
A. ab
B. -ab
C. -a-b
D. 0
E. a/b
Q45. According to the Product Rule of exponents, a^m × a^n equals:
A. a^(mn)
B. a^(m+n)
C. a^(m-n)
D. (2a)^(m+n)
E. a^(m/n)
Q46. According to the Quotient Rule of exponents, for m ≥ n and a ≠ 0, a^m / a^n equals:
A. a^(mn)
B. a^(m+n)
C. a^(m-n)
D. a^(n-m)
E. a^(m/n)
Q47. The Power of a Power rule states that (a^m)^n equals:
A. a^(m*n)
B. a^(m+n)
C. a^(m-n)
D. a^m + a^n
E. a^(n/m)
Q48. The Power of a Product rule states that (ab)^n equals:
A. a*b^n
B. a^n * b
C. a^n * b^n
D. (a+b)^n
E. a^(b*n)
Q49. Transitivity of inequalities states that if a < b and b < c, then:
A. a = c
B. a > c
C. a < c
D. a ≥ c
E. b < a
Q50. The Addition Property of inequalities states that if a < b, then for any c ∈ Z:
A. a+c > b+c
B. a+c < b+c
C. a+c = b+c
D. ac < bc
E. a-c > b-c
Q51. If a < b and c > 0, then multiplying by c gives:
A. ac > bc
B. ac < bc
C. ac = bc
D. a/c > b/c
E. a+c = b+c
Q52. If a < b and c < 0, then multiplying by c gives:
A. ac > bc
B. ac < bc
C. ac = bc
D. -ac < -bc
E. ac ≤ bc
Q53. An integer n is called even if there exists an integer k such that:
A. n = k
B. n = 2k
C. n = 2k + 1
D. n = 3k
E. n = k^2
Q54. An integer n is called odd if it can be expressed as:
A. n = 2k
B. n = 2k + 1
C. n = 3k
D. n = k^2
E. n = k+1
Q55. For integers a and b, we say that a divides b (denoted a | b) if there exists an integer k such that:
A. a = b*k
B. b = a*k
C. k = a*b
D. a = b+k
E. b = a+k
Q56. If a does not divide b, it is denoted as:
A. a / b
B. a != b
C. a ∤ b
D. a \ b
E. a - b
Q57. If a | b and b | c, then according to the transitivity property:
A. b | a
B. c | b
C. a | c
D. c | a
E. a | bc
Q58. If a | b and a | c, then a divides their sum. This is written as:
A. a | (b + c)
B. a | (b * c)
C. a | (b / c)
D. a | (b - c) only
E. a = b + c
Q59. If a | b, then for any integer k:
A. a | k
B. a | kb
C. b | ka
D. k | ab
E. a | b/k
Q60. Which of the following is true for any integer a?
A. a | 0
B. 1 | a
C. a | a
D. All of the above
E. None of the above
Q61. The statement a | 1 holds if and only if:
A. a = 0
B. a = ±1
C. a = 2
D. a > 1
E. a is prime
Q62. If a | b and c | d, then it must be true that:
A. ac | b+d
B. ac | bd
C. a+c | b+d
D. ab | cd
E. b | d
Q63. The statement a | b and b | a hold precisely when:
A. a = 0
B. b = 0
C. a ≠ b
D. a = ±b
E. a > b
Q64. If a | b and b ≠ 0, then the relationship between their absolute values is:
A. |a| > |b|
B. |a| ≤ |b|
C. |a| = |b|
D. |a| ≠ |b|
E. a = b/2
Q65. If a | b and a | c, then a | (bx + cy) for arbitrary integers x and y.
A. True
B. False
C. Only if x = y
D. Only if x=1, y=1
E. Only if a is prime
Q66. In the Division Algorithm (b = aq + r), what is the condition for r?
A. r < 0
B. r > |a|
C. 0 ≤ r < |a|
D. r = |a|
E. 0 < r ≤ a
Q67. In the Division Algorithm equation b = aq + r, what does q represent?
A. Remainder
B. Quotient
C. Divisor
D. Dividend
E. Base
Q68. In the Division Algorithm equation b = aq + r, what does r represent?
A. Remainder
B. Quotient
C. Divisor
D. Dividend
E. Index
Q69. The divisibility condition a | b is true if and only if the remainder r equals:
A. 0
B. 1
C. a
D. -1
E. |a|
Q70. In the Division Algorithm, are the integers q and r unique?
A. Yes, they are unique
B. No, they vary
C. Only for prime numbers
D. Only for positive b
E. Depends on a
Q71. By the Division Algorithm, for 20 = 4q + r, what are the values of q and r?
A. q=4, r=4
B. q=5, r=0
C. q=5, r=1
D. q=4, r=0
E. q=0, r=20
Q72. By the Division Algorithm, for 23 = 5q + r, what are the values of q and r?
A. q=5, r=0
B. q=4, r=2
C. q=4, r=3
D. q=3, r=8
E. q=5, r=-2
Q73. Every even integer n is divisible by 2 because its remainder r when divided by 2 is:
A. 0
B. 1
C. 2
D. -1
E. undefined
Q74. Every odd integer n is not divisible by 2 because its remainder r when divided by 2 is:
A. 0
B. 1
C. 2
D. -1
E. undefined
Q75. A prime number is an integer p > 1 that has exactly:
A. Three divisors
B. One divisor
C. Two distinct positive divisors
D. Four divisors
E. Infinite divisors
Q76. The distinct positive divisors of a prime number p are:
A. 1 and 2
B. 1 and p
C. p and p^2
D. 2 and p
E. 0 and p
Q77. An integer greater than 1 that is not prime is called a:
A. Composite number
B. Even number
C. Odd number
D. Rational number
E. Negative number
Q78. If n is a composite number, then there exist integers a and b such that n = ab, where:
A. a = 1, b > n
B. 1 < a < n, 1 < b < n
C. a = 0, b = n
D. a > n, b > n
E. a = b
Q79. Which of the following is true about the number of prime numbers?
A. There are infinitely many prime numbers
B. There is a finite number of primes
C. Exactly 1000 prime numbers exist
D. They stop after 1,000,000
E. Unknown
Q80. In the sequence N_n = (p1 * p2 * ... * pn) + 1, what is the value of N_1 (where p1=2)?
A. 1
B. 2
C. 3
D. 5
E. 7
Q81. In the sequence N_n = (p1 * p2 * ... * pn) + 1, what is the value of N_2 (p1=2, p2=3)?
A. 5
B. 7
C. 11
D. 13
E. 31
Q82. In the sequence N_n = (p1 * p2 * ... * pn) + 1, what is the value of N_3 ?
A. 17
B. 29
C. 31
D. 41
E. 211
Q83. In the sequence N_n = (p1 * p2 * ... * pn) + 1, what is the value of N_4 ?
A. 31
B. 105
C. 210
D. 211
E. 2311
Q84. In the sequence N_n = (p1 * p2 * ... * pn) + 1, what is the value of N_7 ?
A. 2311
B. 30030
C. 510510
D. 510511
E. 1000000
Q85. Every integer n > 1 can be written uniquely as a product of primes. This is known as the:
A. Fermat's Little Theorem
B. Division Algorithm
C. Fundamental Theorem of Arithmetic
D. Euclidean Algorithm
E. Gauss's Lemma
Q86. Is the number 1 a prime number?
A. Yes, it's prime
B. Yes, it's composite
C. It is both prime and composite
D. It is neither prime nor composite
E. It depends on the context
Q87. Let p be a prime and a, b ∈ N. If p | ab, then:
A. p = a+b
B. p | a AND p | b
C. p | a OR p | b
D. p = ab
E. p = a/b
Q88. If n > 1 is composite, then n has a prime divisor p such that:
A. p > n
B. p = n
C. p ≤ √n
D. p ≥ √n
E. p = n/2
Q89. To check if 97 is prime, we only need to check primes less than or equal to:
A. 5
B. 7
C. 9 (since √97 < 10)
D. 11
E. 13
Q90. Pairs of prime numbers that differ by 4 are called:
A. Twin Primes
B. Cousin Primes
C. Sexy Primes
D. Mersenne Primes
E. Cullen Primes
Q91. What is the Cousin Prime of p = 3 ?
A. 5
B. 7
C. 9
D. 11
E. 13
Q92. What is the Cousin Prime of p = 7 ?
A. 9
B. 10
C. 11
D. 13
E. 15
Q93. What is the Cousin Prime of p = 13 ?
A. 15
B. 17
C. 19
D. 21
E. 23
Q94. A prime number of the form C_n = n(2^n) + 1 is called a:
A. Cousin Prime
B. Mersenne Prime
C. Cullen Prime
D. Fermat Prime
E. Euler Prime
Q95. For n=1, C_1 = 1*(2^1) + 1 = 3. Is C_1 a Cullen Prime?
A. Yes
B. No
C. Only for even n
D. Only for odd n
E. Undefined
Q96. For n=2, C_2 = 2*(2^2) + 1 = 9. Is C_2 a Cullen Prime?
A. Yes
B. No, because 9 is not prime
C. Yes, it's 3^2
D. Yes, for all n
E. Undefined
Q97. For n=3, C_3 = 3*(2^3) + 1 = 25. Is C_3 a Cullen Prime?
A. Yes
B. No, because 25 is not prime
C. Yes, because 25 is odd
D. Yes, it ends in 5
E. Maybe
Q98. A number of the form M_p = 2^p - 1 is called a:
A. Cullen number
B. Mersenne number
C. Cousin number
D. Fermat number
E. Euler number
Q99. What is the value of Mersenne number M_2 (p=2)?
A. 1
B. 2
C. 3
D. 5
E. 7
Q100. What is the value of Mersenne number M_3 (p=3)?
A. 3
B. 5
C. 7
D. 11
E. 15
Q101. What is the value of Mersenne number M_5 (p=5)?
A. 15
B. 25
C. 31
D. 63
E. 127
Q102. If a Mersenne number M_p = 2^p - 1 is prime, then p must necessarily be:
A. Even
B. Composite
C. Prime
D. Negative
E. Perfect square
Q103. Is the condition "p is prime" always sufficient to make M_p = 2^p - 1 a prime number?
A. Yes, always
B. No (e.g., M_11 = 2047 is composite)
C. Only for even p
D. Only for p < 10
E. Yes, proved by Fermat
Q104. If n is a positive composite number, then 2^n - 1 is:
A. Always Prime
B. Always a composite number
C. Always Even
D. A perfect square
E. A power of 2
Q105. The algebraic factorization of (x^n - 1) always contains the factor:
A. x + 1
B. x - 1
C. x^2 - 1
D. x
E. x^n
Q106. Let a > 1 and n > 1. If a^n + 1 is prime, then a must be:
A. Odd
B. Even
C. Prime
D. Composite
E. Zero
Q107. Let a > 1 and n > 1. If a^n + 1 is prime, then n must be of the form:
A. 2k + 1
B. 2^k (a power of 2)
C. 3k
D. p^2
E. n!
Q108. The largest positive integer that divides both a and b without leaving a remainder is called:
A. Least Common Multiple (LCM)
B. Greatest Common Divisor (GCD)
C. Prime factor
D. Quotient
E. Remainder
Q109. What is the value of gcd(a, 0) for any integer a?
A. 0
B. 1
C. |a|
D. a^2
E. Undefined
Q110. Two integers a and b are called co-prime (or relatively prime) if:
A. gcd(a,b) = 0
B. gcd(a,b) = 1
C. a and b are both prime
D. gcd(a,b) = a
E. lcm(a,b) = 1
Q111. Which of the following is equivalent to gcd(a, b)?
A. gcd(a, b-a)
B. gcd(b, a)
C. gcd(a, b+a)
D. gcd(-a, -b)
E. All of the above
Q112. For any integers a, b, and n, the value of gcd(an, bn) equals:
A. n * gcd(a,b)
B. |n| * gcd(a,b)
C. n^2 * gcd(a,b)
D. gcd(a,b)
E. gcd(a,b) / |n|
Q113. If n divides a (n|a) and n divides b (n|b), then n must also divide:
A. a*b
B. gcd(a,b)
C. LCM(a,b)
D. a/b
E. a^b
Q114. For any integers a and b, the gcd(a,b) = d can be written as a linear combination:
A. d = a/b
B. d = a*b
C. d = ax + by
D. d = a-b
E. d = (a+b)/2
Q115. If gcd(a, b) = d, then gcd(a/d, b/d) equals:
A. d
B. 0
C. 1
D. 1/d
E. d^2
Q116. Let a = bq + r. According to the Euclidean Algorithm lemma, gcd(a,b) equals:
A. gcd(a,q)
B. gcd(q,r)
C. gcd(b,r)
D. gcd(a,r)
E. r
Q117. The algorithm that repeatedly applies the division algorithm to find the greatest common divisor is:
A. Fermat's Algorithm
B. RSA Algorithm
C. Euclidean Algorithm
D. Euler's Method
E. Newton's Method
Q118. The Euclidean algorithm process stops when a remainder r_n reaches:
A. 0
B. 1
C. -1
D. 2
E. a
Q119. Using the Euclidean algorithm, what is gcd(75, 45)?
A. 5
B. 10
C. 15
D. 30
E. 45
Q120. Using the Euclidean algorithm, what is gcd(517, 89)?
A. 1
B. 17
C. 89
D. 4
E. 0
Q121. The smallest positive integer that is divisible by both a and b is called:
A. Greatest Common Divisor (GCD)
B. Least Common Multiple (LCM)
C. Prime factor
D. Remainder
E. Quotient
Q122. Which relation connects LCM and GCD for any two integers a and b?
A. LCM = GCD * a * b
B. LCM = a + b - GCD
C. LCM(a,b) * gcd(a,b) = |a * b|
D. LCM / GCD = 1
E. LCM = GCD / |ab|
Q123. If integer a divides b (a|b), then the LCM(a,b) equals:
A. a
B. b
C. ab
D. a/b
E. 1
Q124. What is the Least Common Multiple (LCM) of 12 and 18?
A. 6
B. 24
C. 36
D. 72
E. 216
Q125. A function mapping G × G into G (where a*b ∈ G for all a,b ∈ G) is called a:
A. Unary operation
B. Binary operation
C. Relation
D. Subgroup
E. Isomorphism
Q126. A pair (G, *) consisting of a non-empty set G and a binary operation that satisfies the associative law is called a:
A. Group
B. Semigroup
C. Ring
D. Field
E. Vector space
Q127. Is the pair (N, +) containing Natural numbers and addition considered a semigroup?
A. Yes
B. No
C. Only for primes
D. Only under multiplication
E. Undefined
Q128. For a pair (G, *) to be a "Group", it must satisfy closure, associativity, identity, and:
A. Commutativity
B. Distributivity
C. Inverse element for every element
D. Finiteness
E. Null element
Q129. In any group G, multiplying an element a by its inverse a^-1 always yields:
A. a
B. 0
C. 1
D. e (the identity element)
E. a^-2
Q130. Is the pair (N, +) of Natural numbers and addition considered a Group?
A. Yes
B. No (it lacks inverse elements)
C. Yes, it's Abelian
D. Only for evens
E. None of these
Q131. A group (G, *) that satisfies the commutative law (a*b = b*a) is called a:
A. Normal group
B. Cyclic group
C. Commutative group (Abelian group)
D. Simple group
E. Symmetric group
Q132. The theorem regarding the identity element in any group (G, *) states that it is:
A. Always 0
B. Always 1
C. Unique
D. Infinite
E. Positive
Q133. If e and e' are both identity elements in a group G, then their relation is:
A. e > e'
B. e < e'
C. e = e'
D. e = -e'
E. e * e' = 0
Q134. For a structure to be a Ring (R, +, *), the set R with addition (R, +) must be:
A. A semigroup
B. An Abelian Group
C. A Field
D. A Vector space
E. An empty set
Q135. For a structure to be a Ring (R, +, *), the set R with multiplication (R, *) must be:
A. A Group
B. An Abelian group
C. A Semigroup
D. A Field
E. A Ring
Q136. In a Ring, the laws that connect the operations of addition and multiplication (e.g., a(b+c) = ab+ac) are called:
A. Commutative laws
B. Associative laws
C. Distributive laws
D. Identity laws
E. Inverse laws
Q137. If a Ring satisfies the commutativity of multiplication (a*b = b*a), it is called a:
A. Field
B. Group
C. Commutative ring
D. Semigroup
E. Integral domain
Q138. A Ring containing an element e such that a*e = a for all elements is called a:
A. Simple ring
B. Empty ring
C. Ring with identity
D. Field
E. Matrix ring
Q139. A Field is a Ring where (F, +) is an Abelian group and (F, *) (excluding zero) is also:
A. A Semigroup
B. An Abelian Group
C. A Ring
D. A Module
E. A Lattice
Q140. Does the set of real numbers (R, +, *) constitute a Field?
A. Yes
B. No
C. Only positive reals
D. Only negative reals
E. It is only a ring
Q141. In the group G=Z with operation a*b = a+b+2, what is the identity element e?
A. 0
B. 1
C. -1
D. -2
E. 2
Q142. In the group G=Z with operation a*b = a+b+2, what is the inverse of an element a?
A. -a
B. a+2
C. a-2
D. -a - 4
E. -a + 4
Q143. If (G,*) is a group, then the inverse of a product (a*b)^-1 is equal to:
A. a^-1 * b^-1
B. b^-1 * a^-1
C. a*b
D. -(a*b)
E. e
Q144. We say that a is congruent to b modulo m (a ≡ b mod m) if m divides:
A. a+b
B. a-b
C. ab
D. a/b
E. m-a
Q145. The statement a ≡ b (mod m) holds if and only if there exists an integer k such that:
A. a = b - m
B. a = mk
C. a = b + km
D. b = a + m
E. a = bm
Q146. Why is 25 ≡ 1 (mod 4) a true statement?
A. Because 4 | (25+1)
B. Because 4 | (25*1)
C. Because 4 | (25-1)
D. Because 25 is odd
E. Because remainder is 2
Q147. If n is an even integer, then it is congruent modulo 2 to:
A. 0
B. 1
C. 2
D. -1
E. 3
Q148. If n is an odd integer, then it is congruent modulo 2 to:
A. 0
B. 1
C. 2
D. 3
E. -2
Q149. The property a ≡ a (mod m) represents:
A. Symmetry
B. Reflexivity
C. Transitivity
D. Distributivity
E. Closure
Q150. If a ≡ b (mod m), then b ≡ a (mod m). This represents:
A. Symmetry
B. Reflexivity
C. Transitivity
D. Identity
E. Commutativity
Q151. If a ≡ b (mod m) and b ≡ c (mod m), then a ≡ c (mod m). This represents:
A. Symmetry
B. Reflexivity
C. Transitivity
D. Closure
E. Associativity
Q152. If a ≡ b (mod m) and c ≡ d (mod m), then (a+c) is congruent to:
A. b-d (mod m)
B. b+d (mod m)
C. bd (mod m)
D. b/d (mod m)
E. 0
Q153. If a ≡ b (mod m) and c ≡ d (mod m), then (ac) is congruent to:
A. b+d (mod m)
B. b-d (mod m)
C. bd (mod m)
D. b/d (mod m)
E. 1
Q154. If a ≡ b (mod m), then a^k is congruent to (for any positive integer k):
A. b (mod m)
B. b^k (mod m)
C. k^b (mod m)
D. 1 (mod m)
E. 0
Q155. According to modular arithmetic, (a+b) mod m equals:
A. a mod m + b mod m
B. ((a mod m) + (b mod m)) mod m
C. ab mod m
D. m mod (a+b)
E. 0
Q156. According to modular arithmetic, (ab) mod m equals:
A. a mod m * b mod m
B. ((a mod m) * (b mod m)) mod m
C. a+b mod m
D. m mod (ab)
E. 1
Q157. What is 2008^2008 mod 3?
A. 0
B. 1
C. 2
D. 3
E. 2008
Q158. When calculating factorials modulo 15, any k! where k ≥ 5 is congruent to:
A. 0 (mod 15)
B. 1 (mod 15)
C. 3 (mod 15)
D. 5 (mod 15)
E. 15 (mod 15)
Q159. What is the remainder when (1! + 2! + 3! + 4!) is divided by 15?
A. 0
B. 1
C. 2
D. 3
E. 33
Q160. What is 16 mod 7?
A. 1
B. 2
C. 3
D. 4
E. 5
Q161. What is the remainder when 16^53 is divided by 7?
A. 1
B. 2
C. 3
D. 4
E. 5
Q162. It is 11 PM. What time will it be after 8 hours? (Calculate (11+8) mod 12):
A. 5 AM
B. 6 AM
C. 7 AM
D. 8 AM
E. 9 AM
Q163. What is the least nonnegative residue of 23 modulo 11?
A. 1
B. 2
C. 3
D. 4
E. 5
Q164. What is the least nonnegative residue of 29 modulo 11?
A. 5
B. 6
C. 7
D. 8
E. 9
Q165. What is the least nonnegative residue of 31 modulo 11?
A. 7
B. 8
C. 9
D. 10
E. 0
Q166. What is the least nonnegative residue of 37 modulo 11?
A. 2
B. 3
C. 4
D. 5
E. 6
Q167. What is the least nonnegative residue of 41 modulo 11?
A. 6
B. 7
C. 8
D. 9
E. 10
Q168. What is 10001 mod 10?
A. 0
B. 1
C. 2
D. 9
E. 10
Q169. What is 20000005 mod 10?
A. 0
B. 2
C. 5
D. 7
E. 10
Q170. What is 3004 mod 10?
A. 0
B. 3
C. 4
D. 7
E. 10
Q171. What is the result of (10001 + 20000005 + 3004) mod 10?
A. 0
B. 1
C. 4
D. 5
E. 10
Q172. What is 10001 mod 13?
A. 1
B. 2
C. 3
D. 4
E. 5
Q173. What is 20000005 mod 13?
A. 5
B. 7
C. 9
D. 11
E. 12
Q174. Compute (10001 × 20000005) mod 13:
A. 0
B. 4
C. 8
D. 9
E. 12
Q175. If a ≡ b (mod m) and n divides m (n | m), then it is guaranteed that:
A. a ≡ b (mod n)
B. a ≡ n (mod m)
C. b ≡ m (mod n)
D. a ≡ m (mod n)
E. n ≡ m (mod a)
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