For an integer n, which of the following expressions can never be odd (that is, is always even)?

Introduction / Context:
This question tests understanding of parity, that is, whether expressions are odd or even. We are given several algebraic expressions in terms of an integer n and asked to identify which expression can never be odd. In other words, we must find the expression that is always even regardless of the value of n. This is a fundamental idea in number theory and algebra.

Given Data / Assumptions:

• n is an integer (it may be even or odd).
• We consider the expressions n + 3, 2n, n + 4, 3n, and n - 1.
• We need to find the expression that is always even and therefore never odd.
• We use standard definitions: an even integer is divisible by 2, and an odd integer is of the form 2k + 1 for some integer k.

Concept / Approach:
We analyse the parity of each expression in terms of whether n is even or odd. An even number can be written as 2k, and an odd number as 2k + 1. By substituting these forms into each expression, we determine whether the result is always even, always odd, or can be both depending on n. The expression that is always a multiple of 2, no matter what n is, is the one that can never be odd.

Step-by-Step Solution:
Step 1: Consider the expression 2n. If n is any integer, then 2n = 2 * n, which is always divisible by 2.Step 2: Therefore 2n is always even and can never be odd.Step 3: Now check n + 3. If n = 2k (even), then n + 3 = 2k + 3, which is odd. If n = 2k + 1 (odd), then n + 3 = 2k + 4 = 2(k + 2), which is even. So n + 3 can be odd for some n and even for others.Step 4: Check n + 4. If n = 2k (even), then n + 4 = 2k + 4 = 2(k + 2), which is even. If n = 2k + 1 (odd), then n + 4 = 2k + 5, which is odd. So n + 4 also can be odd for some n.Step 5: Check 3n. If n is even (2k), then 3n = 3 * 2k = 6k, which is even. If n is odd (2k + 1), then 3n = 3(2k + 1) = 6k + 3, which is odd. So 3n can be odd or even depending on n.Step 6: Check n - 1. If n = 2k (even), then n - 1 = 2k - 1, which is odd. If n = 2k + 1 (odd), then n - 1 = 2k, which is even. So n - 1 also can be odd.Step 7: Among all expressions, only 2n is guaranteed to be even for every integer n and can never be odd.

Verification / Alternative check:
You can test specific values of n to verify your conclusions. For example, let n = 2 (even). Then n + 3 = 5 (odd), 2n = 4 (even), n + 4 = 6 (even), 3n = 6 (even), and n - 1 = 1 (odd). Now let n = 3 (odd). Then n + 3 = 6 (even), 2n = 6 (even), n + 4 = 7 (odd), 3n = 9 (odd), and n - 1 = 2 (even). We see that every expression except 2n can switch between odd and even, while 2n always stays even. This confirms our reasoning.

Why Other Options Are Wrong:
Option n + 3: This expression is odd for even n and even for odd n, so it can be odd.Option n + 4: Similarly, it is even for even n and odd for odd n, so it can be odd.Option 3n: When n is odd, 3n is odd; when n is even, 3n is even, so it can be odd.Option n - 1: This expression is odd for even n and even for odd n, so it can be odd as well.

Common Pitfalls:
Students sometimes misinterpret the phrase "cannot be odd" as meaning "is odd", which reverses the logic of the question. Another frequent error is to assume that if an expression is even for some values, then it is always even. Always check both cases, n even and n odd, to understand full parity behaviour. Remember that a coefficient of 2 guarantees that an expression is even, but only if every term has a factor of 2.

Final Answer:
The expression that can never be odd is 2n.