TRB BT Assistant - Euler Function and Congruence Theory Online Test (MCQ Quiz)

1. What is the definition of Euler's totient function, $\phi(n)$?

A) The number of positive divisors of n. B) The number of positive integers less than or equal to n that are relatively prime to n. C) The sum of the positive divisors of n. D) The number of prime factors of n.

2. 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.

3. What is the value of $\phi(10)$?

A) 1 B) 2 C) 4 D) 8

4. What is the value of $\phi(7)$?

A) 1 B) 6 C) 7 D) 0

5. What is the value of $\phi(12)$?

A) 4 B) 6 C) 8 D) 10

6. 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

7. 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 $a^n \equiv 1 \pmod{\phi(n)}$.

8. 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.

9. 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

10. 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 $a/c \equiv b/d \pmod{n}$.

11. What is the remainder when $3^{100}$ is divided by 7?

A) 1 B) 3 C) 5 D) 6

12. What is the remainder when $2^{50}$ is divided by 5?

A) 1 B) 2 C) 3 D) 4

13. What is the value of $\phi(1)$?

A) 0 B) 1 C) Undefined D) Infinite

14. What is the value of $\phi(p)$ where p is a prime number?

A) p B) p-1 C) 1 D) 0

15. What is the result of $5 \pmod{3}$?

A) 1 B) 2 C) 3 D) 5

16. 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

17. What is the multiplicative inverse of 3 modulo 7?

A) 1 B) 2 C) 3 D) 5

18. 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}

19. 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.

20. 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

21. 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

22. What is the value of $\phi(15)$?

A) 1 B) 4 C) 8 D) 12

23. 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

24. 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.

25. 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

26. What is the value of $\phi(8)$?

A) 1 B) 2 C) 4 D) 7

27. What is the result of $17 \pmod{5}$?

A) 1 B) 2 C) 3 D) 4

28. 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

29. What is the value of $\phi(2^3)$?

A) 2 B) 4 C) 6 D) 7

30. Which of the following is equivalent to $10 \pmod{3}$?

A) 0 B) 1 C) 2 D) 3

31. What is the value of $\phi(21)$?

A) 1 B) 10 C) 12 D) 20

32. What is the remainder when $7^{20}$ is divided by 10?

A) 1 B) 3 C) 7 D) 9

33. What is the value of $\phi(30)$?

A) 8 B) 10 C) 12 D) 16

34. What is the remainder when $5^{100}$ is divided by 13?

A) 1 B) 5 C) 8 D) 12

35. 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

36. 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

37. What is the value of $\phi(100)$?

A) 10 B) 20 C) 40 D) 60

38. What is the remainder when $3^{2023}$ is divided by 11?

A) 1 B) 3 C) 5 D) 7

39. What is the multiplicative inverse of 5 modulo 12?

A) 1 B) 5 C) 7 D) 11

40. What is the value of $\phi(99)$?

A) 60 B) 64 C) 70 D) 72

41. 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

42. What is the value of $\phi(10^3)$?

A) 400 B) 500 C) 600 D) 800

43. 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

44. What is the value of $\phi(13)$?

A) 1 B) 12 C) 13 D) 0

45. What is the remainder when $2^{1000}$ is divided by 10?

A) 1 B) 2 C) 4 D) 6

46. What is the value of $\phi(18)$?

A) 1 B) 6 C) 9 D) 12

47. 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

48. 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

49. What is the value of $\phi(1)$?

A) 0 B) 1 C) Undefined D) 2

50. What is the remainder when $11^{100}$ is divided by 7?

A) 1 B) 2 C) 3 D) 4

Euler Function and Congruence Theory - Online Test

Test Results