Write whether every positive integer can be of the form \(4q+2\), where \(q\) is an integer. Justify your answer.
No, not every positive integer can be of the form \(4q+2\).
Step 1: Recall the division algorithm.
When any integer \(n\) is divided by 4, the remainder can only be 0, 1, 2, or 3.
Therefore, every integer can be expressed in one of the following forms:
\(n = 4q\),
\(n = 4q + 1\),
\(n = 4q + 2\),
\(n = 4q + 3\).
Step 2: Check the required form.
The question asks whether every positive integer can be written in the form \(4q + 2\).
But this is only true for those numbers which leave remainder 2 when divided by 4 (like 2, 6, 10, 14, …).
Step 3: Counterexample.
Consider \(n = 5\).
Dividing by 4: \(5 = 4 \times 1 + 1\).
This is of the form \(4q + 1\), not \(4q + 2\).
Conclusion.
Since not all integers fit the form \(4q + 2\), the statement is not true.