Two automated timers are set to chime at regular intervals. The first timer chimes every 1… — GRE General Test

Two automated timers are set to chime at regular intervals. The first timer chimes every $12$ minutes, and the second timer chimes every $18$ minutes. If both timers chime simultaneously at 12:00 PM, how many minutes will pass before they next chime at the exact same time?

Cevap: 36 minutes

Cevap

36 minutes

The time interval between simultaneous chimes is given by the least common multiple (LCM) of the two individual intervals,

12

minutes and

18

minutes. Using prime factorization,

12 = 2^2 \times 3

and

18 = 2 \times 3^2

. The LCM takes the highest power of each prime present:

2^2 \times 3^2 = 4 \times 9 = 36

. Thus,

36

minutes will elapse before the timers chime together again.

Adım Adım Çözüm

Find the prime factorizations of 12 and 18

12 = 2^2 \times 3

and

18 = 2 \times 3^2

Expressing numbers as products of prime factors allows us to easily compute the least common multiple.

Calculate the Least Common Multiple (LCM)

\text{LCM}(12, 18) = 2^{\max(2,1)} \times 3^{\max(1,2)} = 2^2 \times 3^2 = 36

The next simultaneous occurrence of periodic events is given by the LCM of their periods.

Anahtar Kavram

Least Common Multiple (LCM) of periodic events

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