Juxtaposition stands for multiplication on the set of congruence classes or application of the group operation (as applicable)

It was observed that the operations in the cipher (rotation and addition, both on 32-bit words) were somewhat biased over congruence classes mod 3.

In other words, every congruence class except zero modulo p has a multiplicative inverse.

It is a straightforward exercise to show that, under multiplication, the set of congruence classes modulo n which are relatively prime to n satisfy the axioms for an abelian group.

Modulo 1 any two integers are congruent, i.e. there is only one congruence class.

Modulo 2 there is only one relatively prime congruence class, 1, so is the trivial group.

Modulo 4 there are two relatively prime congruence classes, 1 and 3, so the cyclic group with two elements.

A specific partitioning attack called mod n cryptanalysis uses the congruence classes modulo some integer for partitions.

It is also known that in each of the congruence classes 5, 6, 7 (mod 8), for any given k there are infinitely many square-free congruent numbers with k prime factors.

Equivalently, the primes are evenly distributed (asymptotically) among each congruence class modulo d.