Solve the linear congruence 7x+3y\equiv 10(\bmod 16).



Answered question


Solve the linear congruence 7x+3y10(bmod16).

Answer & Explanation



Skilled2021-04-23Added 95 answers

Step 1
Consider the linear congruence 7x+3y10(bmod16).
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+3y10(bmod16) has multiple solutions.
The existence of one solution comes to fruition upon converting the aforementioned linear congruence to the form 7x103ybmod16 and setting y0bmod16.
This leads us to the linear congruence 7x10bmod16.
After multiplying both sides of our congruence by 7, we find x6bmod16.
Therefore, one solution to the linear congruence 7x+3y10(bmod16) is given by
Step 2
Our difference maker comes into play when we let y1bmod16.
This gives rise to the congruence 7x7bmod16.
In this case we have x1bmod16.
As a result, we find another solution of 7x+3y10bmod16 is

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