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Difficulty: MediumHCF and LCM

A digital signal processing unit transmits three distinct telemetry data packets in continuous periodic cycles. The transmission cycles for the three packets take 1415\frac{14}{15} seconds, 75\frac{7}{5} seconds, and 6320\frac{63}{20} seconds, respectively. If all three packets are transmitted simultaneously at the beginning of a diagnostic test, after how many seconds will all three packets be transmitted together again for the first time?

  1. 1265\frac{126}{5}Answer
  2. B
    760\frac{7}{60}
  3. C
    2110\frac{21}{10}
  4. D
    126126

Answer

1265\frac{126}{5}
To determine when all packets will transmit simultaneously again, we calculate the Least Common Multiple (LCM) of their cycle periods. The LCM of fractions is defined as the LCM of their numerators divided by the HCF of their denominators. Given the numerators (14, 7, 63) and denominators (15, 5, 20), we find LCM(14, 7, 63) = 126 and HCF(15, 5, 20) = 5. Dividing these yields the exact time of 1265\frac{126}{5} seconds.

Step-by-Step Solution

1
Identify the mathematical operation required.
The problem asks for the next simultaneous transmission of periodic cycles, which requires finding the Least Common Multiple (LCM) of the given time intervals: 1415\frac{14}{15}, 75\frac{7}{5}, and 6320\frac{63}{20}.
Repeating simultaneous events align at time intervals that are common multiples of their individual periods. The first alignment corresponds to the least common multiple.
2
Apply the rule for finding the LCM of fractions.
The correct mathematical formula is LCM=LCM of numeratorsHCF of denominators\text{LCM} = \frac{\text{LCM of numerators}}{\text{HCF of denominators}}.
This is the required procedure for finding the LCM of rational numbers expressed as simplified fractions.
3
Determine the LCM of the numerators.
The numerators are 1414, 77, and 6363. Their prime factorizations are 14=2×714 = 2 \times 7, 7=77 = 7, and 63=32×763 = 3^2 \times 7. Thus, LCM(14,7,63)=2×32×7=126\text{LCM}(14, 7, 63) = 2 \times 3^2 \times 7 = 126.
The LCM must consist of the highest power of all prime factors present among the numerators.
4
Determine the HCF of the denominators.
The denominators are 1515, 55, and 2020. Their prime factorizations are 15=3×515 = 3 \times 5, 5=55 = 5, and 20=22×520 = 2^2 \times 5. The Highest Common Factor(15,5,20)=5\text{Highest Common Factor}(15, 5, 20) = 5.
The HCF is the largest prime factor shared by all the denominators.
5
Calculate the final LCM of the transmission cycles.
Divide the LCM of the numerators by the HCF of the denominators to obtain exactly 1265\frac{126}{5} seconds.
Applying the values derived in the previous steps to the fraction formula yields the exact coincidence time.

Key Concept

Calculating the LCM of fractions to synchronize periodic events.
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Topics

HCF and LCM

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