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How to find GCD
GCD is found by using all the divisors method listed below.
Find the GCD of 24, 36, and 60
All the divisors of 24 : 1, 2, 3, 4, 6, 8, 12, 24
All the divisors of 36 : 1, 2, 3, 4, 6, 9, 12, 18, 36
Allthe divisors of 60 : 1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30, 60
The common divisors of of 24, 6, 60 : 1,2,3,4,6,12
The greatest of all the common divisors :12
So The G.C.D of 24, 36, 60 is 12
G.C.D by prime factor method
Example: Find G.C.D of 24 and 36 and 60 using prime factor method.
First let us find the prime factors of the given numbers.
24=2x12 36=2x18 60=2x30
24 =2x2x6 36=2x2x9 60=2x2x15
24=2x2x2x3 36=2x2x3x3 60= 2x2x3x5
The factors common to 24,36 and 60 are 2,2,3
Therefore the G.C.D of 24,36and 60 = 2x2x3=12
GCD by Division method
Example : Find GCD of 192 and 120 by division method
120 ) 192 ( 1
-120
72 ) 120 ( 1
-72
48 ) 72 ( 1
-48
24 ) 48 ( 2
- 48
00
The GCD of 192 and 120 is 24
To find L.C.M
To find L.C.M by listing the multiples.
Example : Find the L.C.M of 18 and 12
Multiples of 18 : 18,36,54,72,90,108,126,144.................
Multiples of 12 : 12,24,36,48,60,72,84,96,108 .............
Of the Common Multiples the least is 36
so the L.C.M of 18 and 12 is 36
LCM by prime factor method
To use this method factor each of the numbers into primes. Then for each different prime number in all of the factorizations, do the following...
Count the number of times each prime number appears in each of the factorizations.
For each prime number, take the largest of these counts.
Write down that prime number as many times as you counted for it in step 2.
The least common multiple is the of all the prime numbers written down.
Example: Find the LCM of 18,12, and 9 by prime factor methd.
The prime factor of 18 =2 x 3 x 3
prime factors of 12 = 2 x 2 x 3
Prime factors of 9 = 3 x 3
Factors common to all numbers :3
Factors common to two of the given numbers : 2, 3
Factors not common: 2
L.C.M of 18,12,9 = 3 x 2 x 3 x 2 = 36.
