Solve the linear congruence 7x+3y≡10(mod16)

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Solve the linear congruence  7x+3y≡10(mod16)

Isma Jimenez · Accepted Answer

Step 1 Consider the linear congruence 7x+3y≡10 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+3y≡10 mod 16 has multiple solutions. The existence of one solution comes to fruition upon converting the aforementioned linear congruence to the form 7x≡10−3y mod 16andse∈gy≡0 mod 16. This leads us to the linear congruence 7x≡10 mod 16. After multiplying both sides of our congruence by 7, we find x≡6 mod 16. Therefore, one solution to the linear congruence 7x+3y≡10 mod 16 is given by x≡6 mod 16 y≡0 mod 16 Step 2 Our difference maker comes into play when we let y≡1 mod 16. This gives rise to the congruence 7x≡7 mod 16. In this case we have x≡1 mod 16. As a result, we find another solution of 7x+3y≡10 mod 16 is x≡1 mod 16 y≡1 mod 16

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

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