Euler Function and Congruence Theory - Question Bank

1. What is the value of $\phi(16)$?
A) 2
B) 4
C) 8
D) 15
2. What is the remainder when $11^{100}$ is divided by 7?
A) 1
B) 2
C) 3
D) 4
3. What is the value of $\phi(1)$?
A) 0
B) 1
C) Undefined
D) 2
4. What is the value of $\phi(2^k)$ where k is a positive integer?
A) 2^k
B) 2^k - 1
C) 2^k - 2^{k-1}
D) k
5. What is the solution to the system of congruences $x \equiv 2 \pmod{3}$ and $x \equiv 3 \pmod{5}$?
A) 8
B) 13
C) 17
D) 23
6. What is the value of $\phi(18)$?
A) 1
B) 6
C) 9
D) 12
7. What is the remainder when $2^{1000}$ is divided by 10?
A) 1
B) 2
C) 4
D) 6
8. What is the value of $\phi(13)$?
A) 1
B) 12
C) 13
D) 0
9. Which theorem is related to the primality testing of an integer n by checking if $(n-1)! \equiv -1 \pmod{n}$?
A) Euler's Theorem
B) Fermat's Little Theorem
C) Wilson's Theorem
D) Chinese Remainder Theorem
10. What is the value of $\phi(10^3)$?
A) 400
B) 500
C) 600
D) 800
11. What is the congruence relation for $x^2 \equiv 1 \pmod{8}$?
A) x \equiv 1, 3, 5, 7 \pmod{8}
B) x \equiv 1, 7 \pmod{8}
C) x \equiv 1 \pmod{8}
D) No solution
12. What is the value of $\phi(99)$?
A) 60
B) 64
C) 70
D) 72
13. What is the multiplicative inverse of 5 modulo 12?
A) 1
B) 5
C) 7
D) 11
14. What is the remainder when $3^{2023}$ is divided by 11?
A) 1
B) 3
C) 5
D) 7
15. What is the value of $\phi(100)$?
A) 10
B) 20
C) 40
D) 60
16. What is the value of $\phi(p^k)$ derived from its definition?
A) p^k - 1
B) p^k - p
C) p^k - p^{k-1}
D) p^k
17. If $ax \equiv b \pmod{n}$ has a solution, then it has how many incongruent solutions modulo n?
A) 1
B) gcd(a, n)
C) n/gcd(a, n)
D) n
18. What is the remainder when $5^{100}$ is divided by 13?
A) 1
B) 5
C) 8
D) 12
19. What is the value of $\phi(30)$?
A) 8
B) 10
C) 12
D) 16
20. What is the remainder when $7^{20}$ is divided by 10?
A) 1
B) 3
C) 7
D) 9
21. What is the value of $\phi(21)$?
A) 1
B) 10
C) 12
D) 20
22. Which of the following is equivalent to $10 \pmod{3}$?
A) 0
B) 1
C) 2
D) 3
23. What is the value of $\phi(2^3)$?
A) 2
B) 4
C) 6
D) 7
24. If $a \equiv b \pmod{n}$, then $ka \equiv kb \pmod{n}$ for any integer k. This property is known as:
A) Additivity
B) Multiplicativity
C) Transitivity
D) Scalar Multiplication
25. What is the result of $17 \pmod{5}$?
A) 1
B) 2
C) 3
D) 4
26. What is the value of $\phi(8)$?
A) 1
B) 2
C) 4
D) 7
27. Which theorem states that if p is a prime number, then for any integer a, $a^p \equiv a \pmod{p}$?
A) Euler's Theorem
B) Fermat's Little Theorem
C) Chinese Remainder Theorem
D) Wilson's Theorem
28. What is the order of an element 'a' modulo n?
A) The smallest positive integer k such that $a^k \equiv 1 \pmod{n}$.
B) The number of solutions to $ax \equiv 1 \pmod{n}$.
C) The value of $\phi(n)$.
D) The remainder of a divided by n.
29. If $a \equiv b \pmod{n}$, then $a^k \equiv b^k \pmod{n}$ for any positive integer k. This property is known as:
A) Additivity
B) Multiplicativity
C) Transitivity
D) Power property
30. What is the value of $\phi(15)$?
A) 1
B) 4
C) 8
D) 12
31. What is the value of $\phi(p^2)$ where p is a prime number?
A) p
B) p^2
C) p^2 - p
D) p^2 - 1
32. Consider the system of congruences: $x \equiv 1 \pmod{3}$ and $x \equiv 2 \pmod{5}$. What is a solution for x?
A) 7
B) 11
C) 13
D) 17
33. What is the Chinese Remainder Theorem used for?
A) Solving systems of linear congruences with coprime moduli.
B) Finding the prime factorization of a number.
C) Calculating Euler's totient function.
D) Proving Fermat's Last Theorem.
34. What is the solution to the congruence $2x \equiv 4 \pmod{6}$?
A) x \equiv 2 \pmod{6}
B) x \equiv 4 \pmod{6}
C) x \equiv 2 \pmod{3}
D) x \equiv 1 \pmod{6} and x \equiv 2 \pmod{6}
35. What is the multiplicative inverse of 3 modulo 7?
A) 1
B) 2
C) 3
D) 5
36. If $a \equiv b \pmod{n}$, what can we say about $a$ and $b$?
A) a is a multiple of b
B) b is a multiple of a
C) a and b have the same remainder when divided by n
D) a and b are equal
37. What is the result of $5 \pmod{3}$?
A) 1
B) 2
C) 3
D) 5
38. What is the value of $\phi(p)$ where p is a prime number?
A) p
B) p-1
C) 1
D) 0
39. What is the value of $\phi(1)$?
A) 0
B) 1
C) Undefined
D) Infinite
40. What is the remainder when $2^{50}$ is divided by 5?
A) 1
B) 2
C) 3
D) 4
41. What is the remainder when $3^{100}$ is divided by 7?
A) 1
B) 3
C) 5
D) 6
42. Which of the following is NOT a property of modular arithmetic?
A) If $a \equiv b \pmod{n}$ and $c \equiv d \pmod{n}$, then $a+c \equiv b+d \pmod{n}$.
B) If $a \equiv b \pmod{n}$ and $c \equiv d \pmod{n}$, then $ac \equiv bd \pmod{n}$.
C) If $a \equiv b \pmod{n}$ and $c \equiv d \pmod{n}$, then $a-c \equiv b-d \pmod{n}$.
D) If $a \equiv b \pmod{n}$ and $c \equiv d \pmod{n}$, then $a/c \equiv b/d \pmod{n}$.
43. What does the notation $a \equiv b \pmod{n}$ mean?
A) a divides b
B) n divides (a-b)
C) a and b have the same remainder when divided by n
D) a is congruent to b
44. What is the relationship between Euler's theorem and Fermat's Little Theorem?
A) Fermat's Little Theorem is a special case of Euler's theorem when n is prime.
B) Euler's theorem is a special case of Fermat's Little Theorem.
C) They are unrelated theorems.
D) Euler's theorem applies only to composite numbers, while Fermat's Little Theorem applies to primes.
45. What is Euler's theorem?
A) If gcd(a, n) = 1, then $a^{\phi(n)} \equiv 1 \pmod{n}$.
B) If p is a prime, then $a^p \equiv a \pmod{p}$.
C) If n is composite, then $a^n \equiv a \pmod{n}$.
D) If gcd(a, n) = 1, then $a^n \equiv 1 \pmod{\phi(n)}$.
46. What is the value of $\phi(p^k)$ where p is prime and k is a positive integer?
A) p^k
B) p^k - 1
C) p^k - p^{k-1}
D) k*p
47. What is the value of $\phi(12)$?
A) 4
B) 6
C) 8
D) 10
48. What is the value of $\phi(7)$?
A) 1
B) 6
C) 7
D) 0
49. What is the value of $\phi(10)$?
A) 1
B) 2
C) 4
D) 8
50. Which of the following is NOT a property of Euler's totient function?
A) If p is a prime number, then $\phi(p) = p-1$.
B) If p is a prime number and k is a positive integer, then $\phi(p^k) = p^k - p^{k-1}$.
C) $\phi(mn) = \phi(m)\phi(n)$ for all positive integers m and n.
D) $\phi(mn) = \phi(m)\phi(n)$ if gcd(m, n) = 1.
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