OEF modular arithmetic --- Introduction ---

This module actually contains 23 exercises on computations in the finite ring ZZ/nZZ.

Addition fill

Consider a map , which sends to . Fill the following table for by dragging the numbers given below.


Cubic fill

Consider a map , which sends to . Fill the following table for by dragging the numbers given below.


Division fill

Consider a map , which sends to . Fill the following table for by dragging the numbers given below.


Division I

Compute / in ZZ/ZZ. The result must be represented by a number between 0 and .

Division II

Compute / in ZZ/ZZ. The result must be represented by a number between 0 and .

Division III

Compute / in ZZ/ZZ. The result must be represented by a number between 0 and .

Zero divisors

Is a zero divisor in ZZ/ZZ ?

Zero divisor II

Find the set of zero divisors in ZZ/ZZ. (In this exercise we don't consider 0 as a zero divisor.)

Write each element by a number between 1 and , and separate the elements by commas.


Zero divisors III

We have =2, where is a prime. How many zero divisors there are in ZZ/ZZ ?

In this exercise we don't consider 0 as a zero divisor.


Inverse I

Find the inverse of in ZZ/ZZ. The result must be represented by a number between 0 and .

Inverse II

Find the inverse of in ZZ/ZZ. The result must be represented by a number between 1 and .

Inverse III

Find the inverse of in ZZ/ZZ. The result must be represented by a number between 0 and .

Invertible power

is a prime. Consider the function f: ZZ/ZZ -> ZZ/ZZ defined by f(x)=x .

Is f bijective?


Multiplication fill

Consider a map , which sends to . Fill the following table for by dragging the numbers given below.


Polynomial fill

Consider a map , which sends to . Fill the following table for by dragging the numbers given below.


Powers

Compute the element in ZZ/ZZ. The result must be represented by a number between 0 and .

Powers II

is a prime number. Compute the element in ZZ/ZZ. The result must be represented by a number between 0 and .

Power fill

Consider a map , which sends to . Fill the following table for by dragging the numbers given below.


Roots

is a prime number. There is an element a in ZZ/ZZ, such that a is congruent to modulo . Find a.

The result must be represented by a number between 0 and .


Simple computations modulo n

Compute in ZZ/ZZ. The result must be represented by a number between 0 and .

Squares

Find the set of squares in ZZ/ZZ. (A square in ZZ/ZZ is an element which is the square of another one.)

Write each element by a number between 0 and , and separate the elements by commas.


Sum and product

Find two integers , such that

0 , 0 ,

+ (mod ) , × (mod ) .

You may enter the two numbers in any order.


Trinomial fill

Consider a map , which sends to . Fill the following table for by dragging the numbers given below.

Other exercises on: modular arithmetics  


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Description: collection of exercises on the ring Z/nZ. This is the main site of WIMS (WWW Interactive Multipurpose Server): interactive exercises, online calculators and plotters, mathematical recreation and games

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