The difference between two numbers is 3355. When the larger number is divided by the smaller number, the quotient is 6 and the remainder is 15. What are the smaller number and the larger number respectively?

Introduction / Context:
This question is a classic application of the division algorithm combined with information about the difference between two numbers. It tests your ability to set up and solve linear equations based on quotient, remainder and difference relationships.

Given Data / Assumptions:

• The difference between the two numbers is 3355.
• Larger number divided by smaller number gives quotient 6 and remainder 15.
• We must find both numbers, ordered as (smaller, larger).

Concept / Approach:
Use the division algorithm: if L is the larger number and S is the smaller number, then L = 6S + 15. We also know L - S = 3355. These two equations in two unknowns can be solved simultaneously to find S and L. This is a standard linear equation system problem.

Step-by-Step Solution:
Step 1: Let the larger number be L and the smaller number be S.Step 2: From the quotient and remainder information, write L = 6S + 15.Step 3: From the difference information, write L - S = 3355.Step 4: Substitute L from Step 2 into Step 3: (6S + 15) - S = 3355.Step 5: Simplify: 5S + 15 = 3355.Step 6: Subtract 15 from both sides: 5S = 3340.Step 7: Divide by 5: S = 3340 / 5 = 668.Step 8: Substitute back into L = 6S + 15: L = 6 * 668 + 15 = 4008 + 15 = 4023.

Verification / Alternative check:
Check the difference: 4023 - 668 = 3355, which matches the given difference.Check the division: 4023 divided by 668 gives quotient 6 and remainder 15, because 6 * 668 = 4008 and 4023 - 4008 = 15. This confirms the result.

Why Other Options Are Wrong:
For 546, 3901, the difference is 3355 but 3901 ÷ 546 does not give remainder 15.For 415, 3770, the difference is 3355 but the quotient and remainder are not 6 and 15.For 404, 3759, the difference is 3355, but again 3759 ÷ 404 does not provide the given quotient and remainder.

Common Pitfalls:
Some students may incorrectly set up the division equation as L = 6S - 15 instead of L = 6S + 15.Others may forget to use both conditions simultaneously, checking only the difference or only the division condition.Always translate word statements into algebraic equations carefully and solve them together.

Final Answer:
The smaller number is 668 and the larger number is 4023.