7d \equiv 0 \pmod{12}
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- The time-averaged magnetic flux is:
- Question:** A cultural anthropologist observes a ceremonial circle divided into 12 equal segments, each representing a lunar phase. If a participant starts at segment 1 and jumps \( n \) segments clockwise each day, where \( n \equiv 7 \pmod{12} \), after how many days will the participant return to segment 1?
- The participant jumps \( n = 7 \) segments clockwise each day modulo 12. We seek the smallest positive integer \( d \) such that:
- This means \( 12 \mid 7d \). Since \( \gcd(7, 12) = 1 \), 12 must divide \( d \). Thus, the smallest such \( d \) is:
- d = 12
- After 12 days, the total displacement is \( 7 \times 12 = 84 \equiv 0 \pmod{12} \), returning to the starting segment.
