Question

there are four bells which toll at intervals of 3,7,12,and 14 seconds respectively. the four bella begin to toll at 12 o' clock. when will they next toll together

Answer 1

Four bells toll at 12 'o clock
L.C.M. of 3,7,12,14 =84
84 sec = 1 min 24 sec
so they will toll together after 1min 24 sec.

Answer 2

The four bells will next toll together after 1 minute 24 second

Solution:

Given, there are four bells which toll at intervals of 3, 7, 12, and 14 seconds respectively.  

The four bells begin to toll at 12 o' clock.  

We need to find when will they next toll together.

For that, we have find the L.C.M of all time intervals.

Finding L.C.M of 3, 7, 12, 14

write numbers as multiplication of its prime factors

3 → 3 comes once

7→ 7 comes once

12→ 2 ×3×2 → 2 comes twice and three comes once

14→ 2×7 → 2 comes once and 7 comes once

So the maximum is three comes once and 7 comes once and 2 comes twice

Hence least common multiple is 3×7×2×2 = 84

Now, 84 sec = 1 min 24 sec  

the next toll together after 1 min 24 sec

hence, they will toll together at 12 : 01 : 24