Proof that every integer is of the form 4q, 4q+1, 4q+2 or 4q-1.

Introduction:
In this question, we need to prove that every integer is of the form 4q, 4q+1, 4q+2, or 4q-1, where q is an integer.

Explanation:
To prove this statement, we need to consider the following cases:

Case 1: When the integer is divisible by 4
If an integer is divisible by 4, then we can write it as 4q, where q is an integer.

Case 2: When the integer leaves a remainder of 1 when divided by 4
If an integer leaves a remainder of 1 when divided by 4, then we can write it as 4q+1, where q is an integer.

Case 3: When the integer leaves a remainder of 2 when divided by 4
If an integer leaves a remainder of 2 when divided by 4, then we can write it as 4q+2, where q is an integer.

Case 4: When the integer leaves a remainder of 3 when divided by 4
If an integer leaves a remainder of 3 when divided by 4, then we can write it as 4q-1, where q is an integer.

Conclusion:
Hence, we have proved that every integer is of the form 4q, 4q+1, 4q+2, or 4q-1, where q is an integer.

4q+(4q+1)+(4q+2)= 4q+5q+6q=15q