LCM and HCF
L.C.M by synthetic division method
In order to find the L.C.M of two or more numbers by division method the steps given below are followed.
1) Write all the given numbers in a line seperating them with a comma.
2) Find the prime number which divides at least two of the given numbers
3) And write the Quotient below each of them. carry forward the numbers which are not divisible.
4) Repeat step 2 and 3 until the quotient are coprime to each other.
5) Find the product of all prime divisors along with co-primes in the last column, this product is the L.C.M of the given numbers.
Example I : Find the L.C.M of 24, 36 and 40 by division method.
L.C.M = 2 x 2 x 3 x 2 x 1 x 3 x 5 = 360
Note :
i) If two numbers are relatively primes, then their L.C.M is equal to their product,
Ex : (3,4) =1,
L.C.M = 3 x 4 = 12
ii) In the given numbers if the first number is a multiple of second number then their L.C.M is equal to first number.
Ex : 50 and 25
50 is the multiple of 25
50 is the L.C.M of 50 and 25
iii) The least common multiple of two prime numbers is their product
Ex : LCM of 3 and 5 = 3 x 5= 15
iv) The LCM of two numbers is never less than either of the two numbers.
Ex : LCM of 5 and 6 = 5 x 6 = 30
5 < 30 and 6 < 30
Note : 1. The greatest number that will divide x,y and z leaving remainders a,b,c respectively is given by H.C.F of x-a, y -b, z-c
2. The greatest number that will divide x,y and z leaving the same remainder ‘a’ in each case is given by H.C.F of x-a, y-a, z-a
Example 1:
The HCF of the two numbers is 29 & their sum is 174. What are the numbers?
Solution:
Let the two numbers be 29x and 29y.
Given, 29x + 29y = 174
29(x + y) = 174
x + y = 174/29 = 6
Since x and y are co-primes, therefore, possible combinations would be (1,5), (2,4), (3,3)
For (1,5): 29 x = 29 x 1 and 29 y = 29 (5) = 145
Therefore, the required numbers are 29 and 145.