Five persons fire bullets at a target at an interval of 6, 7, 8, 9 and 12 seconds respectively. The number of times they would fire the bullets…

Correct Option (B)

To determine the number of times multiple events occur simultaneously within a given period, the Least Common Multiple (LCM) of their individual time intervals is calculated.

• The firing intervals for the five persons are 6, 7, 8, 9, and 12 seconds.
• The prime factorization of these intervals is:
  • 6 = 2 × 3
  • 7 = 7
  • 8 = 2³
  • 9 = 3²
  • 12 = 2² × 3
  The LCM is derived by taking the highest power of all prime factors present: 2³ × 3² × 7 = 8 × 9 × 7 = 504 seconds.
• This signifies that all five persons will fire bullets together every 504 seconds.
• The total duration specified is 1 hour, which is equivalent to 3600 seconds.
• To find the number of times they fire together within this hour, the total duration is divided by the LCM:

  $\frac{3600}{504}$5043600​ = 7.14...

• Since only complete instances of simultaneous firing are counted, the integer part of the result, which is 7, represents the number of times they fire together within the hour.

Incorrect Options:

• Options A (6), C (8), and D (9) are incorrect.
• The precise calculation based on the Least Common Multiple of the firing intervals and the total time yields 7 as the number of simultaneous firing instances.
• Option C (8) would be the result if the initial simultaneous firing at the very beginning of the hour (time t=0) were included in the count, in addition to the 7 full cycles calculated. However, based on the provided solution's direct calculation of $\frac{3600}{504}$5043600​ = 7.14... and rounding down to 7, only the full cycles after the initial point are considered.