Proof that Every Integer is of the Form 4q, 4q+1, 4q+2, or 4q+3

Introduction:
In this proof, we will show that every integer can be expressed in one of the forms 4q, 4q+1, 4q+2, or 4q+3, where q is an integer.

Proof:
We will use the division algorithm to prove this statement. According to the division algorithm, given two integers a and b, where b is not zero, there exist unique integers q and r such that a = bq + r, where 0 ≤ r < />

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Case 1: a = 4q
If a is a multiple of 4, then r = 0. Therefore, a can be expressed in the form 4q.

Case 2: a = 4q+1
If a leaves a remainder of 1 upon division by 4, then r = 1. Therefore, a can be expressed in the form 4q+1.

Case 3: a = 4q+2
If a leaves a remainder of 2 upon division by 4, then r = 2. Therefore, a can be expressed in the form 4q+2.

Case 4: a = 4q+3
If a leaves a remainder of 3 upon division by 4, then r = 3. Therefore, a can be expressed in the form 4q+3.

Conclusion:
Therefore, we have shown that every integer can be expressed in one of the forms 4q, 4q+1, 4q+2, or 4q+3, where q is an integer.