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 together at the target in an hour is
- (a) 6
- (b) 7 ✓ UPSC's answer
- (c) 8
- (d) 9
Why the answer is (b)
• The common firing interval is the LCM of 6, 7, 8, 9 and 12 seconds.
• LCM(6, 7, 8, 9, 12) = 2^3 × 3^2 × 7 = 504 seconds.
• One hour is 3600 seconds, and 3600 ÷ 504 = 7 with remainder 672.
• Thus, after the start, they fire together at 504, 1008, 1512, 2016, 2520, 3024 and 3528 seconds.
• Therefore the official key option (b) 7 follows, while counting the initial simultaneous firing at 0 seconds gives the accepted alternate option (c) 8.
Why the other options are wrong
- (a) 6
- Option (a) 6 is too low because 3600 seconds contain 7 full 504-second intervals after the start.
- (c) 8
- Option (c) 8 is the accepted alternate count only if the initial simultaneous firing at 0 seconds is included, not the official key count.
- (d) 9
- Option (d) 9 is too high because 9 × 504 = 4536 seconds, which exceeds 3600 seconds.
Asked in the CSAT Paper II of the UPSC Civil Services Preliminary Examination 2014, held on 24 August 2014. Question and answer key: Union Public Service Commission. Explanation: UPSC Answer Check.