Quantitative - HCF and LCM

HCF and LCM Description

1. Factors & Multiples:- If a/b is completely divisible, then b is a factor of a and a is a multiple of b.
E.g:- 16/4 and 15/5.
4 & 5 are factors of 16 & 15 respectively and 16 & 15 are multiples of 4 & 5.

2. Highest Common Factor (H.C.F.):- Also known as Greatest Common Measure (GCM) or Greatest Common Divisor (GCD), the highest common factor (HCF) of two or more numbers is the greatest number that divides each one of them exactly.
In other words, HCF is the largest factor common to two or more numbers given in a set.

The two methods of finding H.C.F are:-

(i) Factorization Method:- Express each of the given numbers as a product of its most basic prime factors. Choose all the common factors and take the product of these factors which is the HCF of given numbers.
For e.g:- HCF of 36 and 54 is calculated as follows:-


${36 = 2\times 18, 18 = 2\times 9, 9=3\times 3}$
So ${36= 2\times 2\times 3\times 3= 2^{2}\times 3^{2}}$
Now ${54= 2\times 27, 27= 3\times 9, 9 = 3\times 3}$
So ${54 = 2\times 3\times 3\times 3 = 2^{1}\times 3^{3}}$
Now ${2^{1}}$ and ${3^{2}}$ are common to both 36 and 54.
Therefore HCF = ${2^{1}\times 3^{2}}$=18

(ii) Division Method:-
To find the HCF of two numbers, first divide the larger by the smaller number, after that divide the divisor by the remainder. 
Repeat the process of dividing the preceding number by the remainder last obtained till zero is obtained as remainder.
The last divisor is required HCF.
For e.g:- Find the HCF of 1026, 1215 and 2349.

  1. Factors of ${1026 = 2\times 3^{3}\times 19}$
  2. Factors of ${1215 = 3^{5}\times 5 }$
  3. HCF of the two:- ${3^{3} = 27}$
  4. Next step is to find the factors of 2349
  5. ${2349 = 3^{4}\times 29}$
  6. We see what is common between 27 and ${3^{4}\times 29}$, and that is ${3^{3}}$
  7. Hence the HCF of all the three numbers will be 27.
Finding the HCF of more than two numbers:-
Suppose we have to find the HCF of three numbers, then, HCF of [(HCF of any two) and (the third number)] gives the HCF of three given number.
Similarly, the HCF of more than three numbers may be obtained.

3. Least Common Multiple (L.C.M.):- The least number which is exactly divisible by each one of the given number. The two methods of finding L.C.M are:-


(i) Factorization Method:-
Resolve each one of the given numbers into a product of prime factors.
LCM is the product of terms of highest powers of all the factors.
For e.g:- LCM of 7 and 8 is calculated as follows:-

${7 = 7\times 1 = 7^{1}}$
${8 = 2\times 2 = 2^{3}}$
Now LCM is ${7^{1}\times 2^{3} = 56}$

(ii) Division Method (Short Cut):- 
Arrange the given numbers in a row in any order. Divide by a number which divided  exactly at least two of the given numbers and carry forward the numbers which are not divisible.
Repeat the above process till 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.

4. Product of two numbers = Product of their HCF and LCM

5. Co-primes
= Two numbers are said to be co-primes if their HCF is 1.

6. HCF and LCM of Fractions:-
HCF = ${\frac{HCF\quad of\quad Numerators }{LCM\quad of\quad Denominators}}$
   
LCM = ${\frac{LCM\quad of\quad Numerators }{HCF\quad of\quad Denominators}}$

7. HCF and LCM of Decimal Fractions:- To find HCF and LCM of decimal fractions, consider these numbers without decimal point and find HCF or LCM, as the case may be. In the result, mark off as many decimal places as are there in each of the given numbers.
For e.g:- Find the HCF of 1.75, 5.6 and 7

Without decimal point these numbers are 175, 560 and 700
HCF of 175, 560 and 700 is 35.
Hence HCF of 1.75, 5.6 and 7 is 0.35
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