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Zorluk: ZorPrime Factorization, GCD, and LCM

Three automated production lines—Line A, Line B, and Line C—run continuous maintenance cycles every 4545 minutes, 6060 minutes, and 7575 minutes, respectively. All three lines completed a maintenance cycle simultaneously at 8:00 a.m. on Monday. A supervisory inspection is triggered whenever at least two of the three lines complete a maintenance cycle at the exact same time. Between 8:01 a.m. on Monday and 8:00 a.m. on Tuesday, inclusive, how many supervisory inspections will be triggered?

Cevap: 16

Cevap

16
To find the number of times at least two lines finish maintenance simultaneously, we determine the LCM for each pair of intervals (180, 225, and 300 minutes) and calculate how many multiples of these values occur within 1,440 minutes. Within this duration, there are 8 multiples of 180, 6 multiples of 225, and 4 multiples of 300. The single time all three coincide (at 900 minutes) is included in all three pairwise counts. Subtracting 2 instances of this overlap yields 8 + 6 + 4 - 2 = 16 total supervisory inspections.

Adım Adım Çözüm

1
Prime factorize each interval length
45 = 3^2 × 5; 60 = 2^2 × 3 × 5; 75 = 3 × 5^2
Prime factorization allows systematic calculation of pairwise and triple LCMs.
2
Compute pairwise and triple LCMs
lcm(45, 60) = 180; lcm(45, 75) = 225; lcm(60, 75) = 300; lcm(45, 60, 75) = 900
The LCM determines the interval between simultaneous completions.
3
Count occurrences within the 1,440-minute time window
A & B: 8 times; A & C: 6 times; B & C: 4 times; All three: 1 time
Dividing the total time window by each LCM yields the frequency of simultaneous maintenance events.
4
Apply Principle of Inclusion-Exclusion for sets
8 + 6 + 4 - 2(1) = 16 distinct inspection events
Events where all three lines coincide (at t = 900) are counted in all three pair counts and must be adjusted so that the time point is counted exactly once.

Anahtar Kavram

Least Common Multiple (LCM) and Inclusion-Exclusion Principle
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Konular

Prime Factorization, GCD, and LCM

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