Let N be the least positive multiple of 11 that leaves a remainder of 5 when divided by 6, 12, 15, 18. Which one of the following is correct?
- A. 900 \lt N \lt 1000
- B. 1000 \lt N \lt 1100
- C. 1100 \lt N \lt 1200
- D. 1200 \lt N \lt 1300 ✓
Correct Answer: D. 1200 \lt N \lt 1300
Explanation
N = \text{LCM}(6, 12, 15, 18) \times k + 5 = 180k + 5. Since N is a multiple of 11, 180k + 5 \equiv 0 \pmod{11} \implies 4k + 5 \equiv 0 \pmod{11} \implies 4k \equiv 6 \pmod{11}. Multiplying by 3 yields k \equiv 18 \equiv 7 \pmod{11}. For k=7, N = 180(7) + 5 = 1265.
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