Modular Scheduling of Repeating Events
Two maintenance alarms run on cycles. Alarm X sounds every 18 minutes starting at minute 0; alarm Y every 30 minutes starting at minute 0. A technician notes that both alarms fired together at minute 0 and again at some later time, but is unsure when. Separately, a batch process runs every 7 minutes starting at minute 3 (i.e., at minutes 3, 10, 17, …), and the technician wants to know whether it ever coincides with alarm X. The technician suspects that because 7 and 18 share no common factor, coincidence is guaranteed eventually.
1. Find when alarms X and Y next coincide, and the general coincidence times.
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Coincidences occur at multiples of lcm(18,30). 18=2·3², 30=2·3·5, so lcm=2·3²·5=90. They next coincide at minute 90, then every 90 minutes.
2. Determine whether the batch process (minutes 3+7k) ever coincides with alarm X (minutes 18m).
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Need 18m = 3 + 7k, i.e. 18m ≡ 3 (mod 7). 18 ≡ 4 (mod 7), so 4m ≡ 3 (mod 7). Inverse of 4 mod 7 is 2 (since 4·2=8≡1), so m ≡ 6 (mod 7). Smallest m=6 gives minute 108; check 108−3=105=15·7 ✓. So coincidence at minutes 108, 234, … The technician's reasoning is flawed: coprimality of 7 and 18 guarantees a solution to 18m ≡ 3 (mod 7) only because gcd(18,7)=1 divides 3; the guarantee holds here but the reasoning 'no common factor ⇒ guaranteed' needs the divisibility condition gcd(a,n) | b to be valid in general.
3. A third alarm Z runs every 12 minutes from minute 0. Do X, Y, Z ever all coincide after minute 0?
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Triple coincidence requires a multiple of lcm(18,30,12). 12=2²·3, so lcm(18,30,12)=2²·3²·5=180. Yes, at minute 180 and every 180 minutes thereafter.