# 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 statements

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

We first check to see if each ni is pairwise coprime
Since all 0 GCF calculation equal 1, the ni'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 ni in each moduli equation above where x ≡ ai mod ni
N = n1
N = 95x
N = 95

Determine Equation Coefficients denoted as ci
 ci = N ni

Calculate c1
 c1 = 95 95x

c1 = 1

Our equation becomes:
x = a1(c1y1)
x = a1(1y1)
Note: The ai piece is factored out for now and will be used down below

Use Euclid's Extended Algorithm to determine each yi
Using our equation 1 modulus of 95x and our coefficient c1 of 1, calculate y1 in the equation below:
95xx1 + 1y1 = 1
Using the Euclid Extended Algorithm Calculator, we get our y1 = 1

Plug in y values and solve our eqation
x = a1(1y1)
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