A number gives quotient 269 and remainder 0 when divided by 68. When the same number is divided by 67, what is the remainder?

Given data

• N = 68 × 269 (since remainder is 0 on division by 68).
• Find N mod 67.

Concept / Approach

• Use modular arithmetic: (ab) mod m = [(a mod m)(b mod m)] mod m.
• Since 68 ≡ 1 (mod 67), N ≡ 1 × (269 mod 67) (mod 67).

Step-by-step calculation

269 = 67 × 4 + 1 ⇒ 269 mod 67 = 1Therefore, N mod 67 = 1

Verification

N = 68 × 269 = 18,292; 18,292 ÷ 67 = 272 remainder 1.

Common pitfalls

• Reducing both factors needlessly; recognizing 68 ≡ 1 (mod 67) simplifies immediately.

Final Answer

1