# Six bells commence tolling together and toll at intervals of 2, 4, 6, 8, 10, 12, minutes respectively.

Question:

Six bells commence tolling together and toll at intervals of 2, 4, 6, 8, 10, 12, minutes respectively. How many times do they toll together in 30 hours?

Solution:

Six bells toll together at intervals of 2, 4, 6, 8, 10 and 12 minutes, respectively.
Prime factorisation:

$2=2$

$4=2 \times 2$

$6=2 \times 3$

$8=2 \times 2 \times 2$

$10=2 \times 5$

$12=2 \times 2 \times 3$

$\therefore L C M(2,4,6,8,10,12)=2^{3} \times 3 \times 5=120$

Hence, after every 120 minutes (i.e. 2 hours), they will toll together.

$\therefore$ Required number of times $=\left(\frac{30}{2}+1\right)=16$