Albarellak

2021-04-21

Solve the linear congruence $7x+3y\equiv 10\left(b\text{mod}16\right)$ .

AGRFTr

Skilled2021-04-23Added 95 answers

Step 1

Consider the linear congruence$7x+3y\equiv 10\left(b\text{mod}16\right)$ .

Since gcd (7, 3) = 1 we know at least one solution exists.

However, the difference between a linear congruence in one variable and a linear congruence in two variables becomes clear when we see that the congruence$7x+3y\equiv 10\left(b\text{mod}16\right)$ has multiple solutions.

The existence of one solution comes to fruition upon converting the aforementioned linear congruence to the form$7x\equiv 10-3yb\text{mod}16$ and setting $y\equiv 0b\text{mod}16$ .

This leads us to the linear congruence$7x\equiv 10b\text{mod}16$ .

After multiplying both sides of our congruence by 7, we find$x\equiv 6b\text{mod}16$ .

Therefore, one solution to the linear congruence$7x+3y\equiv 10\left(b\text{mod}16\right)$ is given by

$x\equiv 6b\text{mod}16$

$y\equiv 0b\text{mod}16$

Step 2

Our difference maker comes into play when we let$y\equiv 1b\text{mod}16$ .

This gives rise to the congruence$7x\equiv 7b\text{mod}16$ .

In this case we have$x\equiv 1b\text{mod}16$ .

As a result, we find another solution of$7x+3y\equiv 10b\text{mod}16$ is

$x\equiv 1b\text{mod}16$

$y\equiv 1b\text{mod}16$

Consider the linear congruence

Since gcd (7, 3) = 1 we know at least one solution exists.

However, the difference between a linear congruence in one variable and a linear congruence in two variables becomes clear when we see that the congruence

The existence of one solution comes to fruition upon converting the aforementioned linear congruence to the form

This leads us to the linear congruence

After multiplying both sides of our congruence by 7, we find

Therefore, one solution to the linear congruence

Step 2

Our difference maker comes into play when we let

This gives rise to the congruence

In this case we have

As a result, we find another solution of