Which of the following is always odd?

### Concept & Logic The behavior of odd and even numbers under basic arithmetic operations follows strict parity rules. Understanding these foundational rules allows you to eliminate incorrect operations immediately. ### Step-by-Step Solution **Given:** * We need to find the operation between two odd numbers that always results in an odd number. **Calculation / Deduction:** Let's evaluate each arithmetic operation: * **Option (a) Sum:** When you add two odd numbers, the "extra" units pair up to form an even number. (e.g., $3 + 5 = 8$). Always even. * **Option (b) Difference:** Subtracting an odd number from another odd number removes the "oddness", leaving an even result. (e.g., $7 - 3 = 4$). Always even. * **Option (c) Product:** An odd number has no factors of 2. Multiplying two numbers with no factors of 2 results in a product that also has no factors of 2. (e.g., $3 \times 5 = 15$). Always odd. ### Exam Strategy & Shortcut Whenever a question asks "which of the following is always [even/odd]", bypass mental abstractions and quickly plug in small prime numbers like $3$ and $5$. Sum: $3+5=8$ (Even). Difference: $5-3=2$ (Even). Product: $3 \times 5=15$ (Odd). Option (c) is the only one that yields an odd result. This technique takes 3 seconds. ### Common Pitfall A standard trap is overthinking the generalized algebra, such as testing $(2k+1)(2m+1) = 4km + 2k + 2m + 1$, when a simple numerical substitution is far faster and less prone to algebraic errors under exam pressure. ### Final Answer Therefore, the correct answer is **Product of two odd numbers**.