Residue class

Look at

Each number is 6 more than the previous one, and each leaves remainder 1 when divided by 6\.

Now look at

Every one leaves remainder 5 when divided by 6\.

We can therefore sort the integers into repeating families according to their remainder. Such a family is called a \\residue class\\.

Working modulo 6 gives six residue classes: the numbers with remainder 0, the numbers with remainder 1, and so on up to remainder 5\.

The second family above belongs to the residue class 5 modulo 6\. In symbols, its members satisfy

A residue class is not the single number 5\. It is the whole family of integers having that modular position:

The number 5 is simply a convenient \\representative\\ of the class.

More formally, two integers belong to the same residue class modulo $m$ exactly when their difference is divisible by $m$.

(You can think of a residue class as one repeating lane. Moving from one member of the class to the next by adding or subtracting the modulus keeps you in the same lane.)

Residue classes become particularly useful when studying primes.

Modulo 6, every prime number greater than 3 must lie in residue class 1 or residue class 5\. The other four classes have divisibility by 2 or 3 built into them.

But this is only an exclusion rule. It does \\not\\ mean that every number in classes 1 and 5 is prime. For example, 25 lies in class 1 modulo 6, while 35 lies in class 5, and both are composite.

The Prime Spring turns these modular families into visible lanes. Changing the modulus reorganises the same integers into different residue classes, exposing different divisibility patterns without changing the underlying prime numbers.

Residue classes therefore let us replace the question “where is this particular integer?” with a broader one: “which repeating arithmetic family does it belong to?”

See also: modular arithmetic, arithmetic progression, coprime, prime number, Prime Spring.