### HCF and LCM

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Concept -1

HCF → An HCF is a greatest number that divides a series of given numbers exactly.

It stands for highest common factor.

Methods of finding HCF

(a) Breaking into prime factors : - In this method the given numbers are broken into prime numbers. and common from both is taken.

Example → Find HCF of 15 and 35.

Step-I → Break into prime factors.

15 = 3 × 5

35 = 5 × 7

Step -II → Pick up the common multiplicant from both numbers.

So HCF = 5

(b) Methods of division → In this method we divide the greater number by smaller number. and Repeat the procedure by dividing the divisor by remainder.

Process continues till we obtain a zero remainder.

Example → Find HCF of 56 and 70

Step-I →

The Last divisor term will be H.C.F. of numbers. So H.C.F. = 14.

Note → Finding the HCF of more than two numbers by method of division.

Step -I → Find H.C.F. of any two numbers.

Step -II → Then find H.C.F. of H.C.F. and 3rd number. this HCF would be final H.C.F.

Example → Find H.C.F. of 191, 573, 1337

Step-I → Find H.C.F. of 573 and 1337

So 191 is the H.C.F.

Now H.C. F. of 191 and 191 would be = 191.

So H.C.F. of 191

Finding HCF of decimals → Convert them in whole quantity and find H.C.F. of them by above rules.

HCF of fraction → fraction is defined as  =

H.C.F. of fractions =

Example → Find HCF of

H.C.F. = Ans.

Concept-II → LCM

• LCM stands for "Least common multiple"
• It is the least number which is exactly divisible by both or all given numbers.
• Method of finding LCM

Method of prime factors:-

Step-I → Break the number into prime factors.

Step-II → Select the common factors of both the numbers.

Step-III → Then select the remaining factors

Step -IV → Multiply them.

Example → Find LCM of 42 and 70.

42 = 2 × 3 × 7

70 = 2 × 5 × 7

LCM = 2 × 7 × 3 × 5 = 210

LCM of fractions

Example → Find LCM of

Some important formulae.

(a)  LCM × HCF = first number × second number.

(b) If a number is divided by a, b and c. and leaves remainder 'k' in each case then the number would be

LCM of (a, b, c) + K

(c) On dividing a number by 'a', 'b' and 'c' if we get a-k, b-k, and c-k as remainder respectively then that number will be

L.C.M. of (a, b, c) – k.

(d) If a number, after adding 'k' ,is exactly divisible by 'a', 'b' and 'c' then the number will be

L.C.M. of (a, b, c) – k

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