LCM

Chapter - LCM (Lowest Common Mutiple)

Common Multiple - Defined as number which is exactly divisible by each of the numbers under consideration. Ex. 60 is a common multiple of 2,3,4,5,6,10,15,20 and 30 Lowest Common Multiple(LCM): LCM of two or more numbers is the least or smallest number which is exactly divisible by each of them. Ex. Find out LCM of 12, 16 Solution :- Multiples of 12 are 12, 24 ,36, 48, 60 Multiples of 16 are 16, 32, 48, 64 So the least common multiple of both is 48 that is required LCM Methods of Finding LCM 1. Method of Prime Factors Step-1 Resolve each of the given numbers into prime factors Step-2 Find the product of the highest powers of all the factors that occour in the resolution of the given number. This product will be required LCM Ex. Find the LCM of 8, 12, 15 and 21 Solution:- 8= 2*2*2= 2^3  12= 2*2*3= 2^2*3  15=3*5  21=3*7 So Product of prime numbers with highest powers (2^3)*3*5*7=840 required LCM Method 2 : Short Cut Method Write down the given numbers in a line separating them by commas. Divide by any one of prime numbers 2, 3,5,7 which al least exactly divide at least any two of the given numbers . Repeat down the process untill you get a line of prime numbers. Now the product of all divisors and the numbers in the last line will be the required LCM Ex. Find out the LCM of 16, 18, 15, 135, 176 Ex. The traffic lights at three different road crossings change after every 24, 36 and 48 secs respectively. If they all change simultaneously at 09 : 10 : 00 hours , then at what time will they againg change simultaneously ? Solution : Here time requirment is least so we will find LCM Change interval will be LCM of 24, 36, 48 = 144 sec 144 sec= 2 min 24 sec, So interval of change will be at 09 : 12 : 24 hours Ex. The least number, which when divided by 48, 60, 72, 108 and 140 leaves 38, 50, 62, 98 and 130 as remainders respectively, is : Solution :Here (48 - 38 ) = ( 60 - 50 ) = ( 72 - 62 ) = ( 108 - 98 ) = ( 140 - 130 ) = 10 in every case Leat number will be LCM of ( 48, 60, 72, 108, 140 ) - 10 LCM = 15120 So, required number = 15120 - 10 = 15110