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**HCF and LCM**

E.g:- 16/4 and 15/5. 4 & 5 are factors of 16 & 15 respectively and 16 & 15 are multiples of 4 & 5.

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:-

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:-

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.

- Factors of ${1026 = 2\times 3^{3}\times 19}$
- Factors of ${1215 = 3^{5}\times 5 }$
- HCF of the two:- ${3^{3} = 27}$
- Next step is to find the factors of 2349
- ${2349 = 3^{4}\times 29}$
- We see what is common between 27 and ${3^{4}\times 29}$, and that is ${3^{3}}$
- Hence the HCF of all the three numbers will be 27.

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.

(i) Factorization Method:-

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):-

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.

5. Co-primes

HCF = ${\frac{HCF\quad of\quad Numerators }{LCM\quad of\quad Denominators}}$

LCM = ${\frac{LCM\quad of\quad Numerators }{HCF\quad of\quad Denominators}}$

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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