The difference between two numbers is 1365. Dividing the larger by the smaller gives quotient 6 and remainder 15. What is the smaller number?

Problem Restatement

Two numbers (larger L, smaller S) satisfy: L − S = 1365 and L = 6S + 15. Find S.

Given data

• Difference: L − S = 1365
• Division algorithm: L = 6S + 15 (quotient 6, remainder 15)

Concept / Approach

Use simultaneous equations; substitute the division relation into the difference relation.

Step-by-Step Calculation

(6S + 15) − S = 13655S + 15 = 13655S = 1350 → S = 1350 ÷ 5 = 270

Verification

L = 6S + 15 = 6(270) + 15 = 1620 + 15 = 1635Check difference: 1635 − 270 = 1365 ✓Check remainder: 1635 ÷ 270 = 6 remainder 15 ✓

Common Pitfalls

• Forgetting that the remainder must be smaller than the divisor (here 15 < 270).

Final Answer

270