Euler's totient function

Euler's totient function - Quiz

1. What property of Euler's totient function allows phi(m)phi(n) = phi(mn) when gcd(m, n) = 1?

A) Additive
B) Linear
C) Multiplicative
D) Constant

2. Which theorem is used to establish the bijection between A x B and C in the proof of multiplicativity?

A) Euclid's theorem
B) Fundamental theorem of arithmetic
C) Fermat's little theorem
D) Chinese remainder theorem

3. For a prime p and integer k >= 1, what is the value of phi(p^k)?

A) p^k - p
B) p^k - p^k-1
C) p^k-1 - 1
D) p^k - 1

4. Which formula for phi(p^k) is expressed using the product p^k-1(p-1)?

A) p^k-1(p-1)
B) p^k-1(p+1)
C) p^k(p-1)
D) p^k-2(p-1)

5. What is the alternative product formula for phi(p^k) involving 1 - 1/p?

A) p^k(1 + 1/p)
B) p^k-1(1 + 1/p)
C) p^k-1(1 - 1/p)
D) p^k(1 - 1/p)

6. How many multiples of p are there in the set {1, 2, ..., p^k}?

A) p^k-2
B) p
C) p^k-1
D) p^k

7. If gcd(m, n) = 1, what is phi(mn) equivalent to?

A) phi(m) + phi(n)
B) phi(m+n)
C) phi(m) / phi(n)
D) phi(m)phi(n)

8. In the proof for prime power arguments, what are the possible values for gcd(p^k, m)?

A) 1, p, p², ..., p^k
B) 1, p, ..., p^k-1
C) p, p², ..., p^k
D) 1, 2, ..., k

9. The set A is defined as integers coprime to m and less than what value?

A) m
B) mn
C) 1
D) p

10. What is the condition on m for gcd(p^k, m) to be greater than 1?

A) m is coprime to p
B) m is a multiple of p
C) m is a divisor of p
D) m is prime

11. Which of the following is equivalent to Euler's product formula?

A) The square root of n
B) The sum of all prime numbers less than n
C) The product of all integers less than n
D) The formula involving prime factorization exponents

12. In the prime factorization n = p1^k1 * p2^k2... * pr^kr, what are p1, p2, ..., pr?

A) Consecutive integers
B) Even numbers only
C) Any set of positive integers
D) Distinct prime numbers

13. Which formula uses the expression p_i^k_i-1(p_i - 1)?

A) The definition of prime numbers
B) The formula for the sum of divisors
C) The equivalent prime factorization formula
D) The product formula with fractions

14. What is the subject of the provided text regarding formula computation?

A) Probability theory
B) Prime factorization
C) Calculus integration
D) Euler's totient function

15. What are the integers k in the range 1 to n such that gcd(n, k) = 1 called?

A) Multiples
B) Primes
C) Units
D) Totatives

16. What is the value of φ(9)?

A) 9
B) 1
C) 6
D) 3

17. In which cryptographic system is Euler's totient function used?

A) AES encryption system
B) Diffie-Hellman key exchange
C) RSA encryption system
D) DES encryption system

18. For n=9, which numbers are NOT totatives?

A) 3, 6, 9
B) 1, 3, 9
C) 5, 7, 8
D) 1, 2, 4

19. In what year did Leonhard Euler first introduce the totient function?

A) 1879
B) 1784
C) 1801
D) 1763

20. Which Greek letter did Euler use to denote the function in his 1784 publication?

A) sigma
B) phi
C) tau
D) pi

21. Who introduced the now-standard notation φ(A)?

A) Gauss
B) Jordan
C) Euler
D) Sylvester

22. In which work did Gauss introduce the notation for the phi function?

A) The Elements
B) Disquisitiones Arithmeticae
C) Principia Mathematica
D) Arithmetica

23. Who coined the term 'totient' in 1879?

A) Camille Jordan
B) Leonhard Euler
C) Carl Friedrich Gauss
D) J. J. Sylvester

24. How did Gauss write the function notation in his 1801 treatise?

A) πA
B) π(A)
C) φA
D) φ(A)

25. What happens in Euler's product formula when n = 1?

A) It is undefined
B) It equals 1
C) It corresponds to the empty product
D) It equals 0

26. Which principle is suggested as an alternative proof for Euler's product formula?

A) Binomial theorem
B) Inclusion-exclusion principle
C) Mathematical induction
D) Proof by contradiction

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