Introduction:
Homomorphism is a function between two algebraic structures that preserves the operations of the structures. In this problem, we need to find the number of homomorphisms from Z4 to K4.

What is Z4?
Z4 is the group of integers modulo 4. The elements of Z4 are {0, 1, 2, 3} and the group operation is addition modulo 4.

What is K4?
K4 is the complete graph on four vertices. It has four vertices and six edges. K4 is a simple graph, which means that it has no loops or multiple edges.

How to find the number of homomorphisms?
To find the number of homomorphisms from Z4 to K4, we need to consider the possible images of the elements of Z4 under a homomorphism. Since K4 has four vertices, each element of Z4 can be mapped to one of the four vertices.

Case 1: Mapping all elements to the same vertex
If we map all the elements of Z4 to the same vertex in K4, then we get only one homomorphism. This is because the group operation in Z4 is addition modulo 4, and the only way to preserve this operation is to map all the elements to the identity element in K4.

Case 2: Mapping one element to a different vertex
If we map one element of Z4 to a different vertex than the others, then we get four homomorphisms. This is because there are four choices for the vertex that the element is mapped to. Once we have chosen a vertex for the mapped element, then the other elements of Z4 must be mapped to the same vertex to preserve the group operation.

Case 3: Mapping two elements to two different vertices
If we map two elements of Z4 to two different vertices in K4, then we get zero homomorphisms. This is because the group operation in Z4 requires that the sum of any two elements is again an element of Z4. However, if we map two elements to two different vertices in K4, then the sum of those two elements will be mapped to a different vertex, which violates the group operation.

Conclusion:
Therefore, the total number of homomorphisms from Z4 to K4 is 1+4 = 5.