SOFT EARTH EDUCATION

HCF and LCM

 

                            We know that the factors of a number are exact divisors of that particular number. Let’s proceed to the highest common factor (H.C.F.) and the least common multiple (L.C.M.).

 

HCF Definition:

                                 The full form of HCF in Mathematics is Highest Common Factor.

                       As the rules of mathematics dictate, the greatest common divisor or the GCD of two or more positive integers happens to be the largest positive integer that divides the numbers without leaving a remainder. For example, take 8 and 12. The H.C.F. of 8 and 12 will be 4 because the highest number that can divide both 8 and 12 is 4.

​

There are two methods of finding the H.C.F. of a 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.

2. Division Method: Suppose we have to find the H.C.F. of two given numbers. 

   1. Divide the larger number by the smaller one.

   2. Now, divide the divisor by the remainder.

   3. Repeat the process of dividing the receding number by the remainder last obtained till zero is obtained as remainder.

   4. The last divisor is the required H.C.F.

 

Finding the H.C.F. of more than two numbers:

                      Suppose we have to find the H.C.F. of three numbers. Then, H.C.F. of [(H.C.F. of any two) and (the third
number)] gives the H.C.F. of three given numbers.
                      Similarly, the H.C.F. of more than three numbers may be obtained.

​

LCM Definition

                                 The full form of LCM in Mathematics is Least Common Multiple.

                       In arithmetic, the least common multiple or LCM of two numbers say a and b, is denoted as LCM (a, b). And the LCM is the smallest or least positive integer that is divisible by both 'a' and 'b'.

​

There are two methods of finding the H.C.F. of a given set of numbers:

1. Factorization Method of Finding L.C.M.: Resolve each one of the given numbers into a product of prime factors. Then, L.C.M. is the product of highest powers of all the factors.

2. Common Division Method {Short-cut Method) of Finding L.C.M.:

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

   2. Divide by a number which divides exactly at least two of the given numbers and carry forward the numbers which are not divisible.

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

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

​

HCF and LCM Formula

                    The formula which involves both HCF and LCM is:

Product of Two numbers = (HCF of the two numbers) x (LCM of the two numbers)

​

Factors and Multiples:

                       If a number a divides another number b exactly, we say that a is a factor of b. In this case, b is called a multiple of a.

​

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

 

                                1. H.C.F. = --------------------------------------------------------------------------

                                 

                                2. L.C.M. = --------------------------------------------------------------------------

​

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

                     In given numbers, make the same number of decimal places by annexing zeros in some numbers, if necessary. Considering these numbers without decimal point, find H.C.F. or L.C.M. as the case may be. Now, in the result, mark off as many decimal places as are there in each of the given numbers.

Comparison of Fractions:

                    Find the L.C.M. of the denominators of the given fractions. Convert each of the fractions into an equivalent fraction with L.C.M. as the denominator, by multiplying both the numerator and denominator by the same number.