**Equivalence Relation**

To show that R is an equivalence relation on the set of integers Z, we need to demonstrate that it satisfies three properties: reflexivity, symmetry, and transitivity.

**Reflexivity**

Reflexivity means that every element of Z is related to itself. In this case, we need to show that for every integer x, (x, x) is in R.

Let's take an arbitrary integer x. Since any integer minus itself is always zero, x - x = 0. Now, we need to check if 0 is a multiple of a non-zero integer 5. Since 0 can be written as 5 * 0, we can see that 0 is indeed a multiple of 5. Therefore, (x, x) is in R for any integer x, and R is reflexive.

**Symmetry**

Symmetry means that if (x, y) is in R, then (y, x) must also be in R.

Let's assume that (x, y) is in R, which means x - y is a multiple of 5. Now, we need to show that (y, x) is also in R, implying that y - x is a multiple of 5.

Since x - y is a multiple of 5, we can write it as 5 * k for some integer k. If we multiply both sides of the equation by -1, we get -x + y = -5 * k, which can be rewritten as y - x = 5 * (-k). Here, (-k) is also an integer, so y - x is a multiple of 5. Therefore, (y, x) is in R whenever (x, y) is in R, and R is symmetric.

**Transitivity**

Transitivity means that if (x, y) and (y, z) are in R, then (x, z) must also be in R.

Assume that (x, y) and (y, z) are in R, which means x - y and y - z are multiples of 5, denoted as 5 * m and 5 * n, respectively.

To prove transitivity, we need to show that (x, z) is in R, implying that x - z is a multiple of 5.

By substituting the values of x - y and y - z, we have x - z = (x - y) + (y - z) = 5 * m + 5 * n = 5 * (m + n). Here, (m + n) is an integer, so x - z is a multiple of 5. Therefore, (x, z) is in R whenever (x, y) and (y, z) are in R, and R is transitive.

**Conclusion**

Since R satisfies reflexivity, symmetry, and transitivity, it is an equivalence relation on the set of integers Z.