#
OEF modular arithmetic
--- Introduction ---

This module actually contains 23 exercises on computations in the
finite ring /*n*.

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

### Division II

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

### Division III

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

### Zero divisors

Is a zero divisor in / ?

### Zero divisor II

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

### Inverse I

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

### Inverse II

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

### Inverse III

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

### Invertible power

is a prime. Consider the function f: / -> / 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 /. The result must be represented by a number between 0 and .

### Powers II

is a prime number. Compute the element ^{} in /. 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 /, 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 /. The result must be represented by a number between 0 and .

### Squares

Find the set of squares in /. (A square in / 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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