Addition fill

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

$(val10[$m_i])

Cubic fill

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

$(val10[$m_i])

Division fill

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

$(val10[$m_i])

Division I

Compute $val10/$val9 in $m_ZZ/$val7$m_ZZ. The result must be represented by a number between 0 and $val8.

Division II

Compute $val10/$val9 in $m_ZZ/$val7$m_ZZ. The result must be represented by a number between 0 and $val8.

Division III

Compute $val10/$val9 in $m_ZZ/$val7$m_ZZ. The result must be represented by a number between 0 and $val8.

Zero divisors

Is $val7 a zero divisor in $m_ZZ/$val6$m_ZZ ?

Zero divisor II

Find the set of zero divisors in $m_ZZ/$val6$m_ZZ. (In this exercise we don't consider 0 as a zero divisor.)

Write each element by a number between 1 and $val7, and separate the elements by commas.


Zero divisors III

We have $val7=$val62, where $val6 is a prime. How many zero divisors there are in $m_ZZ/$val7$m_ZZ ?

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


Inverse I

Find the inverse of $val9 in $m_ZZ/$val7$m_ZZ. The result must be represented by a number between 0 and $val8.

Inverse II

Find the inverse of $val9 in $m_ZZ/$val7$m_ZZ. The result must be represented by a number between 1 and $val8.

Inverse III

Find the inverse of $val9 in $m_ZZ/$val7$m_ZZ. The result must be represented by a number between 0 and $val8.

Invertible power

$val9 is a prime. Consider the function f: $m_ZZ/$val9$m_ZZ -> $m_ZZ/$val9$m_ZZ defined by f(x)=x$val14 .

Is f bijective?


Multiplication fill

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

$(val10[$m_i])

Polynomial fill

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

$(val15[$m_i])

Powers

Compute the element $val9$val7 in $m_ZZ/$val6$m_ZZ. The result must be represented by a number between 0 and $val8.

Powers II

$val6 is a prime number. Compute the element $val9$val7 in $m_ZZ/$val6$m_ZZ. The result must be represented by a number between 0 and $val8.

Power fill

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

$(val10[$m_i])

Roots

$val6 is a prime number. There is an element a in $m_ZZ/$val6$m_ZZ, such that a$val10 is congruent to $val16 modulo $val6. Find a.

The result must be represented by a number between 0 and $val7.


Simple computations modulo n

Compute $val13 in $m_ZZ/$val6$m_ZZ. The result must be represented by a number between 0 and $val7.

Squares

Find the set of squares in $m_ZZ/$val6$m_ZZ. (A square in $m_ZZ/$val6$m_ZZ is an element which is the square of another one.)

Write each element by a number between 0 and $val7, and separate the elements by commas.


Sum and product

Find two integers $val7, $val8 such that

0 $val7 $val10 , 0 $val8 $val10 ,

$val7 + $val8 $val13 (mod $val9) , $val7 × $val8 $val14 (mod $val9) .

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.

$(val12[$m_i])