Question

Three bells commenced to toll at the same time and tolled at intervals of \[20,{\text{ }}30,{\text{ }}40{\text{ }}seconds\] respectively. If they toll together at\[6am\], then which of the following is the time at which they can roll together.

Answer 1

Hint:As in the above question, it is asked at what time all the bells toll together. We have to find the Lowest Common Multiple that is L.C.M. to get the desired result. The L.C.M. of three numbers is the smallest number which is divisible by all the three numbers and is not equal to zero. The LCM of three numbers can be obtained by division method.

Complete step by step answer:
We have been given three bells that commence tolling together and toll at intervals of \[20,{\text{ }}30,{\text{ }}and{\text{ }}40{\text{ }}seconds\] respectively. We will get the time interval at which all the three bells will toll together by taking the L.C.M. or Least Common Multiple of all the three intervals. So, we have to find the L.C.M. of \[20,{\text{ }}30,{\text{ }}and{\text{ }}40\]. This is done by the division method.

The numbers will be operated simultaneously. The factors in the division method must divide the minimum of two numbers and it should be until the last term of the factorization becomes either \[1\]or a prime number to be divided. Then we will multiply all the prime factors in order to get the result. So for the L.C.M. of \[20,{\text{ }}30,{\text{ }}40\] the prime factorization will be represented as follows.
\[
  2\left| {\underline {\,
  {20,\,\,30,\,40} \,}} \right. \\
  5\left| {\underline {\,
  {10,15,\,20} \,}} \right. \\
  2\left| {\underline {\,
  {2,\,\,3,\,\,4} \,}} \right. \\
  \,\,\,\,1,\,\,3,\,\,2 \\
 \]
Now for finding the L.C.M. we will multiply the divisor and the prime numbers which are left behind. So, the L.C.M is
\[2 \times 5 \times 2 \times 1 \times 3 \times 2 = 120\]
So the bells will toll together after\[120{\text{ }}seconds\].
Now, \[60\,seconds = 1\,min\]
So \[120\,seconds = 60 \times 2 = 2\,minutes\]
Therefore, all the three bells will toll together\[2\,minutes\] after \[6:00\,am\].

Hence they will toll together at \[6:02\,am\].

Note:While dealing with application related H.C.F., L.C.M. problems it must be carefully observed what the difference between them is and which one of them is needed to apply. H.C.F. is the highest number which divides all the numbers whereas L.C.M. is the least common multiple of the given numbers.