Remainder computation — When the number 121012 is divided by 12, what remainder results? Use modular properties for a quick solution.

Introduction / Context:
Dividing by 12 benefits from splitting into factors 3 and 4 and using modular rules. This avoids long division and speeds up competitive exams and interviews.

Given Data / Assumptions:

• Dividend: 121012.
• Divisor: 12 = 3 * 4.
• Goal: remainder r where 0 ≤ r < 12.

Concept / Approach:
Chinese Remainder reasoning: compute the number mod 3 and mod 4, then find the unique r in 0..11 satisfying both conditions. Use digit-sum for mod 3 and last-two-digits rule for mod 4.

Step-by-Step Solution:
Compute mod 3: sum of digits = 1+2+1+0+1+2 = 7 → 7 mod 3 = 1.Compute mod 4: last two digits = 12 → 12 mod 4 = 0.Find r such that r ≡ 1 (mod 3) and r ≡ 0 (mod 4).Candidates for mod 4 are 0, 4, 8. Check mod 3: 0→0, 4→1, 8→2. Thus r = 4.

Verification / Alternative check:
Direct modulo: 121012 = 121000 + 12; since 121000 mod 12 = 4 (because 120000 is divisible by 12 and 1000 mod 12 = 4), then 4 + 12 mod 12 = 4, confirming r = 4.

Why Other Options Are Wrong:

• 0 / 2 / 3 / 6: Do not satisfy both congruences r ≡ 1 (mod 3) and r ≡ 0 (mod 4).

Common Pitfalls:
Adding digits incorrectly; forgetting the last-two-digits rule for mod 4; assuming r must be the mod 3 remainder itself.

Final Answer:
4