 # Solve the linear congruence 7x+3y -= 10(mod 16) Phoebe 2021-02-21 Answered
Solve the linear congruence
$7x+3y\equiv 10\left(\text{mod}16\right)$
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Step 1
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 has multiple solutions.
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 is given by

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 is