Consider the numbers 12 and 15 :
The multiples of 12 are : 12, 24, 36, 48, 60, 72, 84, ....
The multiples of 15 are : 15, 30, 45, 60, 75, 90, ....
60 is a common multiple (a multiple of both 12 and 15), and there are no lower common multiples.
Therefore, the lowest common multiple of 12 and 15 is 60.
Steps to Find LCM:
1. Find prime factorisation of given numbers
2. Take the product of all distinct prime numbers with greatest power.
Eg: LCM of 36, 40 and 72Step1: prime factorisation of
36=22x32
40=23x51
72=23x32
Step 2: Take product of all prime numbers with highest power
23x32x51 = 360
therefore, lcm of (36,40 and 72) is 360
Note:
Lcm of fractions is ratio of lcm of numerators to hcf of denominatorsThe lcm of given numbers is always greater than or equal to highest number in the given numbers.
Product of two numbers is equal to product of their lcm and hcf.
Lcm is always a multiple of hcf.
What is the least number which when divided by 8,9,12 and 15 leaves the same remainder 1 in each case
soulution :Required number = (l.c.m. of 8, 9, 12, 15) + 1 = 361
An electronic device makes a beep after every 60 sec. Another device makes a beep after every 62 sec. They beeped together at 10 a.m. The time when they will next make a beep together at the earliest, is
Solution: L.C.M. of 60 and 62 seconds is 1860 sec = 31 min.
They will beep together at 10.31 a.m.
The least number of five digits which is exactly divisible by 12, 15 and 18 is
Solution:Least number of 5 digits is 10000.
L.C.M. of 12, 15, 18 is 180.
On dividing 10000 by 180, the remainder is 100.
Required number = 10000 + (180 - 100) = 10080.
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
Solution:L.C.M. of 2, 4, 6, 8, 10, 12 is 120.
So, the bells will toll together after every 120 seconds i.e. 2 minutes.
In 30 minutes, they will toll together in 30/2 + 1 = 16 times.