A number divided by 68 gives the quotient 260 and remainder zero. If the same number is divided by 65, the remainder is

### Concept & Logic This problem applies the Division Algorithm and modular arithmetic properties. The Division Algorithm states: $Dividend = Divisor \times Quotient + Remainder$. When evaluating the remainder of a product, if any individual factor of that product is perfectly divisible by the new divisor, the entire product is perfectly divisible (leaving a remainder of $0$). ### Step-by-Step Solution * **Given:** A number $N$ when divided by $68$ gives a quotient of $260$ and remainder $0$. * **Step 1: Express the number as an equation.** $N = 68 \times 260 + 0$ $N = 68 \times 260$ * **Step 2: Test divisibility by the new divisor (65).** We need to find the remainder of $(68 \times 260) \div 65$. Check the factors $68$ and $260$ to see if they relate to $65$. $68 \div 65$ leaves a remainder of $3$. $260 \div 65$ leaves a remainder of $0$ (since $65 \times 4 = 260$). * **Step 3: Calculate the final remainder.** Since $260$ is a multiple of $65$, the entire product $(68 \times 260)$ is a multiple of $65$. Therefore, the remainder is $0$. ### Exam Strategy & Shortcut Never multiply the divisor and quotient out (e.g., doing $68 \times 260$). Immediately check if the new divisor ($65$) is a factor of either the old divisor ($68$) or the quotient ($260$). Noticing that $260$ is exactly $4$ times $65$ makes the answer instantly $0$. ### Common Pitfall A major time-waster is multiplying $68 \times 260 = 17,680$ and then performing long division by $65$. This introduces massive room for error on a question designed to take seconds. ### Final Answer **Therefore, the correct answer is 0.**