Difficulty: Medium
Correct Answer: 6
Explanation:
Introduction:
This aptitude question tests your skill in modular arithmetic and remainder calculations. Instead of performing a huge multiplication, you can simplify each factor modulo 8 and then combine the results.
Given Data / Assumptions:
Concept / Approach:
When working modulo 8, we can replace each number by its remainder when divided by 8. Then we multiply these remainders and finally reduce the product modulo 8. Formally: (a * b * c) mod 8 = [(a mod 8) * (b mod 8) * (c mod 8)] mod 8. This greatly simplifies calculations.
Step-by-Step Solution:
Step 1: Find each factor modulo 8. Compute 1234 mod 8: 8 * 154 = 1232, remainder 1234 - 1232 = 2. So 1234 ≡ 2 (mod 8).
Compute 1235 mod 8: 1235 = 1232 + 3, so remainder is 3. So 1235 ≡ 3 (mod 8).
Compute 1237 mod 8: 1232 + 5 = 1237, so remainder is 5. So 1237 ≡ 5 (mod 8).
Step 2: Multiply the remainders. (1234 × 1235 × 1237) mod 8 ≡ (2 × 3 × 5) mod 8. Compute: 2 × 3 = 6. 6 × 5 = 30. Now find 30 mod 8: 8 * 3 = 24, remainder 30 - 24 = 6. Thus: (1234 × 1235 × 1237) mod 8 = 6.
Verification / Alternative Check:
We used only modulo arithmetic, which preserves remainders under multiplication. The intermediate remainders 2, 3, and 5 are correct because each is found by subtracting the nearest multiple of 8. The final remainder calculation 30 mod 8 = 6 is also straightforward.
Why Other Options Are Wrong:
Remainders 0, 1, 2, or 4 would correspond to different modulo products such as 2 * 4 = 8, 2 * 2 = 4, etc. None matches the computed result 6 when the actual remainders (2, 3, and 5) are used correctly.
Common Pitfalls:
Errors often arise from miscomputing the remainders of the original numbers or from forgetting to reduce after each multiplication. Another pitfall is trying to multiply the large numbers directly and then dividing, which is time-consuming and prone to calculation mistakes.
Final Answer:
The remainder when 1234 × 1235 × 1237 is divided by 8 is 6.
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