HCF and LCM

To check the definition of HCF and LCM, look into the page http://arithmophobia-elixir.blogspot.in/2014/07/basic-facts-and-formula.html

Methods of Finding HCF:

1. Factorization Method : Express each one of the given numbers as the product of prime factors. The product of least powers of common prime factors gives HCF.

Example: Find HCF of 108, 288, 360

108 = 2² * 3³ , 288 = 2⁵ * 3², 360 = 2³ * 3² * 5

We can also write it as : 108 = 2² * 3³ * 5⁰ , 288 = 2⁵ * 3² * 5⁰, 360 = 2³ * 3² * 5¹ ( as 5⁰ = 1 and 5¹ = 5 )

Here, we have put 5⁰ just for understanding and to make all the factors 2, 3 and 5 common among all three numbers.

Here, We have expressed 108, 288 and 360 as a product of prime factors.

We need to break all the factors until it cannot be broken further (so as to reach the prime factor).

After factorizing we see the common factors of 108, 288 and 360 are 2, 3 and 5. Now, the least power for 2 is 2, for 3 is 2 and for 5 is 0 among all three given numbers.

Hence, HCF = 2² * 3² * 5⁰ = 4 * 9 * 1 = 36

2. Division Method: Suppose we have to find the HCF of two given numbers. Divide the larger number by the smaller one. Now divide the divisor by the remainder . Repeat the process of dividing the preceeding number by the remainder last obtained until zero is obtained in the remainder. The last divisor is the required HCF.

Note: To find the HCF of more than two numbers. Find the HCF of any two number. And then take that HCF and another number from the given numbers and again find the HCF of those two. Keep repeating this until at the end you get one HCF.

Suppose we have to find the HCF of three given numbers.

Then HCF of [(HCF of any two) and ( the third number)] gives the HCF of three numbers.

Example: Find HCF of 513, 1134, 1215

        ____

^{1134 ) 1215 ( 1}

            ¹¹³⁴

_____{______}

^{81 ) 1134 ( 14}

          ⁸¹

^__________

        ³²⁴

        ³²⁴

_______

          ⁰

_______

HCF of 1134 and 1215 is 81. ( If we have to find the HCF of 1134 and 1215 then 81 is the answer)

Here we have to find the HCF of three numbers 513, 1134 and 1215.

Hence required HCF = HCF of 513 and 81.

      ___

^{81 ) 513 ( 6}

        ⁴⁸⁶

_____{__}

^{27 ) 81 ( 3}

        ⁸¹

^__________

       ⁰

_______

We see the last divisor is 27.

Hence, HCF of required numbers is 27.

Methods of Finding LCM:

1. Factorization method: Express each one of the given numbers as the product of prime factors. The product of highest powers of common prime factors gives LCM.

Example: Find LCM of 108, 288, 360

108 = 2² * 3³ , 288 = 2⁵ * 3², 360 = 2³ * 3² * 5

We can also write it as : 108 = 2² * 3³ * 5⁰ , 288 = 2⁵ * 3² * 5⁰, 360 = 2³ * 3² * 5¹ ( as 5⁰ = 1 and 5¹ = 5 )

Here, we have put 5⁰ just for understanding and to make all the factors 2, 3 and 5 common among all three numbers.

Here, We have expressed 108, 288 and 360 as a product of prime factors.

We need to break all the factors until it cannot be broken further (so as to reach the prime factor).

After factorizing we see the common factors of 108, 288 and 360 are 2, 3 and 5. Now the highest power for 2 is 5, for 3 is 3 and for 5 is 1 among all three given numbers.

Hence, LCM = 2⁵ * 3³ * 5 = 32 * 27 * 5 = 4320

2. Common Division Method: Arrange the given numbers in a row in any order. Divide by a number which divides exactly at least two of the given numbers and carry forward the numbers which are not divisible. Repeat the above process until no two of the numbers are divisible by the same number except 1. The product of the divisors and the undivided numbers is the required LCM of the given numbers.

Example: Find LCM of 513, 1134 and 1215

3	513	1134	1215
3	171	378	405
3	57	126	135
3	19	42	45
	19	14	15

Here divisors are 4 times 3 at the left and undivided numbers are 19, 14 and 15 from the bottom of the above table.

Hence required LCM = 3 * 3 * 3 * 3 * 19 * 14 * 15 = 323190

Note: We have taken same set of numbers for both LCM and HCF to make you understand the difference between LCM and HCF. We will solve more questions on this later. Please let us know if you have any doubts, by posting your comments/suggestions either on facebook ( www.facebook.com/mathselixir) or here at below. You need to login with your gmail ID or open ID to post the comments.