7 bells start tolling together and toll at intervals of 3, 6, 9, 12, 15, 18 and 21 seconds respectively. In 2½ hours, how many times do they toll together?

7 bells start tolling together and toll at intervals of 3, 6, 9, 12, 15, 18 and 21 seconds respectively. In 2½ hours, how many times do they toll together? Correct Answer 8

The L.C.M of 3, 6, 9, 12, 15, 18 and 21 is 1260

Hence, the bell will toll together after every 1260 seconds that is after every 21 minutes

2 ½ hours or 150 minutes

First they will toll together at the first minute

Then after 21 minutes i.e. 1+ 21 = 22^nd minute, (22 + 21) 43^rd minute, and last at 148^th minute.

Considering above as arithmetic progression where 1 is the first and 148 is the last n^th term,

⇒ 1, 22, 43, …, 148

⇒ T_n = 1 + (n – 1)21

⇒ 148 = 1 + 21n – 21

⇒ 168 = 21n

⇒ n = 8

∴ They will together 8 times in duration of 150 minute