Question

If $q$ is an integer, then which of the following is a positive odd integer?

• A. $4q + 1$
• B. $4q$
• C. $4q + 2$
• D. $4q + 4$

Hint:

If a base expression is even (like $4q$), adding an odd number always flips parity to odd.

Answer 1

The Correct Option is A

Concept: To determine whether an algebraic expression is odd or even, we use basic number theory:
• An even integer is of the form $2n$
• An odd integer is of the form $2n + 1$
• Adding an even number does not change parity
• Adding 1 to an even number always produces an odd number We analyze each option carefully step by step.

Step 1: Analyze Option (2) $4q$
Since $4q = 2(2q)$, it is always divisible by 2 for any integer $q$. Therefore, it is always even, never odd.

Step 2: Analyze Option (3) $4q + 2$
We rewrite: $$4q + 2 = 2(2q + 1)$$ This clearly shows it is divisible by 2, so it is always even.

Step 3: Analyze Option (4) $4q + 4$
We factor: $$4q + 4 = 4(q + 1) = 2(2q + 2)$$ This is also always divisible by 2, hence always even.

Step 4: Analyze Option (1) $4q + 1$
We know: $$4q = \text{even number}$$ Adding 1: $$\text{even} + 1 = \text{odd}$$ So $4q + 1$ is always odd for any integer $q$.

Step 5: Positivity check
For $q \ge 0$, the smallest value is: $$4(0) + 1 = 1$$ which is positive. For larger integers, the expression increases further and remains positive. Thus, $4q + 1$ is a positive odd integer. Final Conclusion: Only $4q + 1$ satisfies both conditions: being odd and positive.