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Euler's totient function - Quiz
Contributed by: Herbert
  • 1. What property of Euler's totient function allows phi(m)phi(n) = phi(mn) when gcd(m, n) = 1?
A) Additive
B) Constant
C) Multiplicative
D) Linear
  • 2. Which theorem is used to establish the bijection between A x B and C in the proof of multiplicativity?
A) Chinese remainder theorem
B) Fundamental theorem of arithmetic
C) Fermat's little theorem
D) Euclid's theorem
  • 3. For a prime p and integer k >= 1, what is the value of phi(pk)?
A) pk-1 - 1
B) pk - p
C) pk - pk-1
D) pk - 1
  • 4. Which formula for phi(pk) is expressed using the product pk-1(p-1)?
A) pk-1(p-1)
B) pk-1(p+1)
C) pk(p-1)
D) pk-2(p-1)
  • 5. What is the alternative product formula for phi(pk) involving 1 - 1/p?
A) pk-1(1 - 1/p)
B) pk(1 - 1/p)
C) pk-1(1 + 1/p)
D) pk(1 + 1/p)
  • 6. How many multiples of p are there in the set {1, 2, ..., pk}?
A) pk-1
B) p
C) pk-2
D) pk
  • 7. If gcd(m, n) = 1, what is phi(mn) equivalent to?
A) phi(m) + phi(n)
B) phi(m) / phi(n)
C) phi(m)phi(n)
D) phi(m+n)
  • 8. In the proof for prime power arguments, what are the possible values for gcd(pk, m)?
A) 1, 2, ..., k
B) 1, p, p2, ..., pk
C) p, p2, ..., pk
D) 1, p, ..., pk-1
  • 9. The set A is defined as integers coprime to m and less than what value?
A) m
B) 1
C) p
D) mn
  • 10. What is the condition on m for gcd(pk, m) to be greater than 1?
A) m is a multiple of p
B) m is a divisor of p
C) m is coprime to 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 product of all integers less than n
C) The formula involving prime factorization exponents
D) The sum of all prime numbers less than n
  • 12. In the prime factorization n = p1k1 * p2k2... * prkr, what are p1, p2, ..., pr?
A) Consecutive integers
B) Distinct prime numbers
C) Any set of positive integers
D) Even numbers only
  • 13. Which formula uses the expression p_ik_i-1(p_i - 1)?
A) The formula for the sum of divisors
B) The definition of prime numbers
C) The product formula with fractions
D) The equivalent prime factorization formula
  • 14. What is the subject of the provided text regarding formula computation?
A) Calculus integration
B) Euler's totient function
C) Prime factorization
D) Probability theory
  • 15. What are the integers k in the range 1 to n such that gcd(n, k) = 1 called?
A) Totatives
B) Units
C) Primes
D) Multiples
  • 16. What is the value of φ(9)?
A) 1
B) 3
C) 9
D) 6
  • 17. In which cryptographic system is Euler's totient function used?
A) DES encryption system
B) AES encryption system
C) Diffie-Hellman key exchange
D) RSA encryption system
  • 18. For n=9, which numbers are NOT totatives?
A) 1, 2, 4
B) 3, 6, 9
C) 1, 3, 9
D) 5, 7, 8
  • 19. In what year did Leonhard Euler first introduce the totient function?
A) 1763
B) 1784
C) 1879
D) 1801
  • 20. Which Greek letter did Euler use to denote the function in his 1784 publication?
A) phi
B) tau
C) sigma
D) pi
  • 21. Who introduced the now-standard notation φ(A)?
A) Gauss
B) Euler
C) Sylvester
D) Jordan
  • 22. In which work did Gauss introduce the notation for the phi function?
A) Arithmetica
B) Disquisitiones Arithmeticae
C) The Elements
D) Principia Mathematica
  • 23. Who coined the term 'totient' in 1879?
A) J. J. Sylvester
B) Carl Friedrich Gauss
C) Leonhard Euler
D) Camille Jordan
  • 24. How did Gauss write the function notation in his 1801 treatise?
A) π(A)
B) φA
C) πA
D) φ(A)
  • 25. What happens in Euler's product formula when n = 1?
A) It equals 0
B) It corresponds to the empty product
C) It equals 1
D) It is undefined
  • 26. Which principle is suggested as an alternative proof for Euler's product formula?
A) Proof by contradiction
B) Mathematical induction
C) Binomial theorem
D) Inclusion-exclusion principle
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