The difference between two numbers is 1365. When the larger number is divided by the smaller one, the quotient is 6 and the remainder is 15. The smaller number is

## Concept & Formula This problem utilizes the fundamental Division Algorithm rule, combined with a simple linear equation for the difference. $$ \text{Dividend} = (\text{Divisor} \times \text{Quotient}) + \text{Remainder} $$ ## Step-by-Step Solution * **Given:** Let the larger number be $x$ (Dividend) and the smaller number be $y$ (Divisor). Difference: $x - y = 1365$ Quotient $= 6$, Remainder $= 15$ * **Calculation / Deduction:** * Apply the division rule: $x = (y \times 6) + 15$ * $x = 6y + 15$ * Substitute this expression for $x$ into the difference equation: * $(6y + 15) - y = 1365$ * Combine like terms: * $5y + 15 = 1365$ * Subtract 15 from both sides: * $5y = 1350$ * Divide by 5: * $y = 270$ * The question asks for the smaller number, which is $y$. ## Exam Strategy & Shortcut **Back-Solving via Options:** We know the larger number is $x = y + 1365$. We also know $x = 6y + 15$. This means $y + 1365 = 6y + 15$. Visually isolate $y$: $5y = 1350$. Divide $1350$ by $5$ mentally (double it to $2700$, drop a zero $= 270$). This avoids writing out the full formal steps. ## Common Pitfall A common error is setting up the division equation backwards, writing $y = 6x + 15$, which will yield a negative or nonsensical fractional answer. Always clearly label your variables as "larger" and "smaller" to ensure the dividend and divisor are placed correctly. ## Final Answer **Therefore, the correct answer is 270.**