Question

Six bells commence tolling together and toll at intervals of 2, 4, 6, 8, 10 and 12 seconds respectively. In 30 minutes, how many times do they toll together?
1) 4
2) 10
3) 15
4) 16

Answer 1

Hint: In the above given problem, we will find the time of the tolling of the bells together by calculating the LCM of the given time seconds at different intervals.
LCM is the lowest common factor, which is found out by expanding the given integers in their multiples or we can say prime factors.
Using the method of LCM we will calculate the value of time of tolling of the bells together.

Complete step-by-step solution:
Let's discuss the LCM and calculation method of the above problem in more detail first and then we will calculate the value of time.
First we will expand all the given values of intervals and then take the common values out and multiply then remaining values with the common values.
$ \Rightarrow 2 = 1 \times 2$
$ \Rightarrow 4 = 2 \times 2$
$ \Rightarrow 6 = 3 \times 2$
$ \Rightarrow 8 = 2 \times 2 \times 2$
$ \Rightarrow 10 = 5 \times 2$
$ \Rightarrow 12 = 2 \times 2 \times 3$
From the given expansion we find that 2 is common to all the values;
$\therefore 2 \times 2 \times 2 \times 3 \times 5 = 120$
We got 120 seconds.
In 30 minutes the bells will toll together;
120 seconds means 2 minutes, this means that after every 2 minutes the bells will toll together;
If we write 0, 2, 4, 6, 8, 10, 12, 14, 16,18, 20, 22, 24, 26, 28, 30 then after counting we get 16 times the bells will toll after every 2 minutes together.

Option 4 is correct.

Note: LCM has another method to be calculated which is a grid method, in which we use to write the all the numbers in a grid table all together and then keep on dividing the all numbers by selecting a common multiple for all numbers in order to get 1 at the end of the calculation.