Three automated data backup protocols in a cloud server execute network synchronization pu… — State PSC Exam

Three automated data backup protocols in a cloud server execute network synchronization pulses at precise intervals. Protocol Alpha pulses every $\frac{18}{5}$ seconds, Protocol Beta every $\frac{24}{25}$ seconds, and Protocol Gamma every $\frac{36}{35}$ seconds. They all initiate their first pulse simultaneously at 08:00:00 AM. Assuming the initial pulse at 08:00:00 AM is excluded from the count, which of the following represents the total number of times all three protocols will pulse together in exactly $12$ minutes?

A
$21000$
B
$10$
$50$ Cevap
D
$51$

Cevap

The data protocols will pulse together

50

times in the

12

-minute period.

To find when all three protocols pulse together, we must determine the Least Common Multiple (LCM) of their cycle times. The LCM of fractions is calculated by dividing the LCM of their numerators (

18, 24, 36

) by the HCF of their denominators (

5, 25, 35

). The LCM of

18, 24,

and

36

72

. The HCF of

5, 25,

and

35

5

. This gives a synchronized interval of

72/5 = 14.4

seconds. In

12

minutes (

720

seconds), the number of synchronized intervals is

720 \div 14.4 = 50

. Since the initial pulse is excluded as per the instructions, the correct answer remains

50

Adım Adım Çözüm

Identify the mathematical concept required to find when all three protocols pulse simultaneously.

The time of simultaneous pulsing is the Least Common Multiple (LCM) of their individual cycle times:

\frac{18}{5}

\frac{24}{25}

, and

\frac{36}{35}

Simultaneous occurrence of periodic events always aligns at multiples common to all individual intervals.

Apply the formula for finding the LCM of fractions.

\text{LCM of fractions} = \frac{\text{LCM of numerators}}{\text{HCF of denominators}}

This is the standard algebraic rule to find the least common multiple for fractional values.

Calculate the LCM of the numerators (

18, 24, 36

) and the HCF of the denominators (

5, 25, 35

\text{LCM}(18, 24, 36) = 72

and

\text{HCF}(5, 25, 35) = 5

. Thus, the synchronized interval is

\frac{72}{5}

seconds, or

14.4

seconds.

Prime factorization gives

18 = 2 \times 3^2

24 = 2^3 \times 3

36 = 2^2 \times 3^2

(LCM =

2^3 \times 3^2 = 72

). For denominators,

5

is the largest common divisor.

Convert the given total time window into seconds.

12 \text{ minutes} = 12 \times 60 = 720 \text{ seconds}

Units must match the cycle interval (seconds) before division.

Calculate the total number of simultaneous pulses.

720 \div 14.4 = 50

. Since the initial pulse is excluded, the final count remains

50

Dividing the total time window by the synchronized interval yields the number of subsequent events.

Anahtar Kavram

Least Common Multiple (LCM) of fractions and its application to periodic simultaneous events.

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