Miscellaneous Concepts of Number System

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Question 1

Euler's theorem states: IF a and n are coprime, them a ϕ(n) ≡ ? mod  n

  • ϕ(n)

  • 1

  • a

  • 0

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Question 2

Which of the following is not a valid use of Euler's theorem?

  • To compute large powers modulo a number

  • When base and modulus are not coprime

  • To reduce exponent using φ(n)

  • To speed up modular exponentiation

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Question 3

What is φ(10), where φ is the euler's totient function?

  • 5

  • 6

  • 4

  • 7

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Question 4

If a and m are coprime, then [Tex]a^{-1} ≡ a^{ϕ(m) −1} \mod m[/Tex] is derived from:

  • Fermat’s Little Theorem

  • Euler’s Theorem

  • Wilson’s Theorem

  • CRT

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Question 5

Project Euler problem solving often uses which trick to compute big mod powers efficiently?

  • Simple multiplication

  • Brute force

  • Modular exponentiation

  • Binary search

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Question 6

Which of the following is NOT required to apply Fermat's Little Theorem?

  • Modulus must be prime

  • Base must be coprime with modulus

  • Base must be even

  • Exponent can be reduced mod (p−1)

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Question 7

Fermat's theorem is used to compute which of the following efficiently?


  • Primality testing

  • Modular inverses

  • Fast exponentiation

  • All of the above

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Question 8

What is 316 mod  17 using FLT?

  • 1

  • 3

  • 0

  • 2

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Question 9

Why is FLT insufficient for deterministic primality testing?

  • Fails for Carmichael numbers

  • Only works for even numbers

  • Only works for natural numbers

  • None of the above

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Question 10

What is the remainder when 5100 is divided by 13?

  • 1

  • 5

  • 8

  • 12

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There are 10 questions to complete.

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