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

Introduction / Context:
This is a number theory problem involving division with remainder and the difference between two numbers. We are told the difference between the larger and smaller numbers and given the quotient and remainder when the larger number is divided by the smaller. The goal is to find the smaller number. Such problems appear frequently in quantitative aptitude tests and require careful use of the division algorithm.

Given Data / Assumptions:
- There are two numbers, a larger one and a smaller one.
- The difference between the larger and smaller numbers is 1365.
- When the larger number is divided by the smaller number, the quotient is 6 and the remainder is 15.
- We need to find the smaller number.

Concept / Approach:
The division algorithm states that if a larger number L is divided by a smaller number S, then L = S * quotient + remainder. Here, the quotient is 6 and the remainder is 15, so L = 6S + 15. We also know that the difference L - S equals 1365. Substituting the expression for L into the difference equation allows us to solve for S directly.

Step-by-Step Solution:
Step 1: Let the smaller number be S and the larger number be L.Step 2: From the division condition, L = 6S + 15.Step 3: The difference between the larger and smaller numbers is given as L - S = 1365.Step 4: Substitute L = 6S + 15 into the difference equation: (6S + 15) - S = 1365.Step 5: Simplify the left side: 6S + 15 - S = 5S + 15.Step 6: So we get 5S + 15 = 1365.Step 7: Subtract 15 from both sides to get 5S = 1350.Step 8: Divide both sides by 5: S = 1350 / 5 = 270.Step 9: Therefore, the smaller number is 270.

Verification / Alternative check:
Compute the larger number using L = 6S + 15. With S = 270, L = 6 * 270 + 15 = 1620 + 15 = 1635. The difference L - S is 1635 - 270 = 1365, which matches the given difference. Also, when 1635 is divided by 270, the quotient is 6 and the remainder is 15, since 270 * 6 = 1620 and 1635 - 1620 = 15. This confirms that the smaller number is indeed 270.

Why Other Options Are Wrong:
If S were 280, the larger number would be 6 * 280 + 15 = 1695, and the difference would be 1695 - 280 = 1415, not 1365. Similar contradictions arise with S = 370 or S = 170 or S = 260. None of those values give both the correct difference and the required quotient and remainder when the larger number is divided by the smaller.

Common Pitfalls:
Some learners misapply the division formula and write L = S / 6 + 15 or similar incorrect expressions. Others mistakenly set the difference as S - L instead of L - S. To avoid these errors, remember that L = S * quotient + remainder and that the difference is larger minus smaller. Writing these relationships clearly before solving makes the algebra straightforward.

Final Answer:
The smaller number is 270.