If a is any integer then, for any positive integer, n, we can write a= qn+ r for a and r integers and0\le r< n. That is the "Euclidean division algorithm" which basically says that if you divide a by n you get some quotient, q, and remainder r.
"x is congruent to y modulo (not "module") n" if and only if x- y is divisible by n. By the Euclidean division algorithm, we can write x= q_1n+ r_1 and y= q_2n+ r_2 for some integers q_1, q_2, r_1, and r_2. Then x- y]= q_1n+ r_1- (q_2n+ r_2)= (q_1- q_2)n+ (r_1- r_2). Since r_1 and r_2 are both non-negative and less than n, there difference is between -n and n. Since x- y is divisible by n, we must have r_1- r_2= 0 or r_1= r_2. If x and y are equivalent modulo n (if x- y is divisible by n) then they have the same remainder when divided by n.
That is, we can label the equivalence classes (set of numbers that are all equivalent to one another modulo n) by the remainders when the numbers in that equivalence class are divided by n. Given any number, x, its remainder, when divided by n, is an integer from 0 to n-1. Every integer is in exactly one of those classes because it has exactly one such remainder.
