Using the Chinese Remainder Theorem, solve the following system of modulo equations: x = 9 mod 95 x = 0 mod 97 x = 30 mod 98 x = 55 mod 99

Enter modulo statementsUsing the Chinese Remainder Theorem, solve the following system of modulo equations x ≡ 9 mod 95x

We first check to see if each n_{i} is pairwise coprime Since all 0 GCF calculation equal 1, the n_{i}'s are pairwise coprime, so we can use the regular formula for the CRT

Calculate the moduli product N We do this by taking the product of each n_{i} in each moduli equation above where x ≡ a_{i} mod n_{i} N = n_{1} N = 95x N = 95

Determine Equation Coefficients denoted as c_{i}

c_{i} =

N

n_{i}

Calculate c_{1}

c_{1} =

95

95x

c_{1} = 1

Our equation becomes: x = a_{1}(c_{1}y_{1}) x = a_{1}(1y_{1}) Note: The a_{i} piece is factored out for now and will be used down below

Use Euclid's Extended Algorithm to determine each y_{i} Using our equation 1 modulus of 95x and our coefficient c_{1} of 1, calculate y_{1} in the equation below: 95xx_{1} + 1y_{1} = 1 Using the Euclid Extended Algorithm Calculator, we get our y_{1} = 1

Plug in y values and solve our eqation x = a_{1}(1y_{1}) x = 9 x 1 x 1 x = 9 x = 9

Now plug in 9 into our 1 modulus equations and confirm our answer Equation 1: 9 ≡ 9 mod 95x We see from our multiplication lesson that 95x x 0 = 0 Adding our remainder of 9 to 0 gives us 9