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. What is the smaller number?

### Concept & Formula This problem requires translating two distinct conditions into a system of linear equations using the standard Division Algorithm. $$ \text{Dividend} = (\text{Divisor} \times \text{Quotient}) + \text{Remainder} $$ ### Step-by-Step Solution * **Given:** Let the larger number be $L$ and the smaller number be $S$. * Condition 1: The difference is 1365. $L - S = 1365$ — (Equation 1) * Condition 2: Division algorithm variables. Dividend = $L$, Divisor = $S$, Quotient = 6, Remainder = 15. $L = 6S + 15$ — (Equation 2) * **Calculation:** * Substitute the value of $L$ from Equation 2 into Equation 1: $(6S + 15) - S = 1365$ * Combine like terms: $5S + 15 = 1365$ * Isolate the variable term: $5S = 1365 - 15$ $5S = 1350$ * Solve for the smaller number: $S = 1350 / 5$ $S = 270$ ### Exam Strategy & Shortcut Once you establish the equation $5S = 1350$, look at the options before doing full division. The number 1350 ends in a 0. When divided by 5, the result must end in an even number or zero. $135 \div 5 = 27$, so appending the zero gives 270. It quickly matches option (b). ### Common Pitfall Students sometimes reverse the variables in the difference equation (writing $S - L = 1365$), leading to negative numbers and confusion. Always explicitly define which variable represents the larger value before building equations. ### Final Answer Therefore, the correct answer is **270**.