Solve the linear congruence 7x+3y -= 10(mod 16)

Phoebe 2021-02-21 Answered
Solve the linear congruence
7x+3y10(mod16)
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Expert Answer

Isma Jimenez
Answered 2021-02-22 Author has 84 answers

Step 1
Consider the linear congruence 7x+3y10 mod 16.
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 mod 16 has multiple solutions.
The existence of one solution comes to fruition upon converting the aforementioned linear congruence to
the form 7x103y mod 16andsegy0 mod 16.
This leads us to the linear congruence 7x10 mod 16.
After multiplying both sides of our congruence by 7, we find x6 mod 16. Therefore, one solution to the linear congruence 7x+3y10 mod 16 is given by
x6 mod 16
y0 mod 16
Step 2
Our difference maker comes into play when we let y1 mod 16.
This gives rise to the congruence 7x7 mod 16.
In this case we have x1 mod 16.
As a result, we find another solution of 7x+3y10 mod 16 is
x1 mod 16
y1 mod 16

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