HOW TO SOLVE HCF AND LCM PROBLEMS

About the topic "How to solve hcf and lcm problems" 

"How to solve hcf and lcm problems ?" is a big question having had by the people who get prepared for competitive exams and study quantitative aptitude. For some students, solving problems on hcf and lcm is never being easy and always it is a challenging one.

How to solve hcf and lcm  problems ?

The answer  for the question "How to solve hcf and lcm problems ?" is purely depending upon the question that we have in the topic "H.C.F and L.C.M". The techniques and methods we apply to solve  problems on hcf and lcm  will vary from problem to problem.

The techniques and methods we apply to solve a particular  problem  will not work for another problem on HCF and LCM.

Even though we have different techniques to solve problems in different topics of math, let us see "How to solve hcf and lcm  problems"

Methods to find H.C.F of the given numbers

Highest Common Factor (H.C.F) or Greatest Common Divisor (G.C.D):

                          The H.C.F of two or more numbers is the greatest number that divides each of them exactly.

Methods to find H.C.F:

There are two methods to find H.C.F of given set of numbers

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 H.C.F            

Example :

Find the H.C.F of 108, 288 and 360.

Let us see, how H.C.F can be found for the given three numbers.

First let us write the given numbers as the product of prime factors.

108 = 2²X3³

288 = 2⁵X3²

360 = 2³X5X3²

When we look in to the prime factors of the given numbers, we find 2 and  3 in common of all the three numbers. The least power of 2 is 2 and 3 is also 2.

Now, to find the H.C.F, we just have to multiply 2² and 3².

Hence, the H.C.F = 36

2. Division Method: 

                            To find the H.C.F of two given numbers using division method, please follow the following steps.

Step 1 : Divide the larger number by the smaller one. You will get some remainder

Step 2 : Now, divide the divisor (smaller one in the above) by the remainder of step 1.When you do so, again, you will get some remainder. 

Step 3 : Again you have to divide the divisor (remainder of step1) by the remainder of step 2.

We have to continue the same process, until we get the remainder zero. 

It has been clearly shown in the following HCF and LCM problems.

Example :

Find the H.C.F of 1134 and 1215.

In the above example, the larger number is 1215 and the smaller number is 1134. As we explained above,we do the following steps.

Step 1:

We divide the lager number 1215 by the smaller number 1134. When we do so, we get the remainder 81.

Step 2:

Now we divide the divisor (1215) by the remainder of step 1(that is 81).

Step 3:

On continuing this process, we get remainder zero when we divide the divisor of step1 (that is 1215) by the remainder of step2 (that is 81).

Here the H.C.F is 81. Because we get the remainder zero when we divide by 81.

Hence, the H.C.F = 81

From the above example, it is very clear that H.C.F is nothing but the divisor for which we get the remainder is zero.

How to find H.C.F for more than two numbers:  

If we want to find H.C.F of three numbers, first find H.C.F of any two numbers. Then, H.C.F of [H.C.F of two numbers and the third number] gives H.C.F of three numbers. In the same manner, H.C.F of more than three numbers may be obtained..

On this webpage of HCF and LCM problems, next, we are going to see L.C.M

Least Common Multiple (L.C.M):

               The least number which is exactly divisible by  each one of the given numbers is called their L.C.M.

1. Factorization Method:

                   Resolve each one of the given numbers in to a product of prime factors. Then, L.C.M is the product of highest powers of all the factors.            

Example:

Find the L.C.M of 108, 288 and 360.

Let us see, how L.C.M can be found for the given three numbers.

First let us write the given numbers as the product of prime factors.

108 = 2²X3³

288 = 2⁵X3²

360 = 2³X5X3²

The prime factors we find above are 2,3 and 5. The highest power of 2 is 5, 3 is 3 and 5 is 1

To find L.C.M, we just have to multiply 2⁵, 3³ and 5

Hence, the L.C.M = 4320

2. Common Division Method (Short-cut Method) :

                         To find the L.C.M of two given numbers using division method, please follow the following steps.

Step 1  : Arrange the given numbers in a row in any order.

Step 2 : Divide by a number which divides exactly at least two of the given numbers and write the remaining numbers as it is.

Step 3 :Repeat the same process till no two of the numbers are divisible by the same number except 1.

Step 4 :The product of the divisors and undivided numbers is the required L.C.M of the given numbers.   

It has been clearly shown in the following example.

Example:

Find the L.C.M of 16, 24, 36 and 54

In the above calculation, the divisors are 2,2,2,3,3 and undivided numbers are 2,1,13. To find the L.C.M, we just have to multiply the divisors and undivided numbers.

Product = 2X2X2X3X3X2X1X1X3 = 432

Hence, the L.C.M is 432.

Co.Primes :

                 If the H.C.F of two numbers is 1, they are called as co-primes. 

Example:

                  3 and 4 co-primes.Because, the H.C.F of 3 and 4 is 1  That is,  there is no common factor between them except 1.

H.C.F and L.C.M of Fractions :

When students have a look on the above examples and steps explained, they will get answer for the question "How to solve hcf and lcm problems?"  We hope, hereafter they will not search answer for the question "How to solve hcf and lcm problems?"

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