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LCM and HCF

Formulae :
HCF LCM = First number Second Number

HCF of Fraction = lcm

LCM of Fraction =lcm2

Least number :
Least number exactly divisible by a,b and c = LCM of a,b and c
Least number divided by a,b and c leaves remainder x,y and z =(LCM of a,b and c)- P ,
where P= a-x = b-y = c-z.

Greatest number :
Greatest No. exactly divisible by a,b and c = HCF of a,b and c
Greatest No. divided by a,b and c leaves remainder x,y and z(HCF of a,b and c)- P , where P= a-x = b-y = c-z.

For Example- Find the L.C.M. of 72, 240, 196.
Explanation : Using Prime factorisation method

72 = 2×2×2×3×3 = 2³×3²

240 = 2×2×2×2×3×5 = 24×3×5

196 = 2×2×7×7 = 2²×7²

L.C.M. of the given numbers = product of all the prime factors of each of the
given number with greatest index of common prime factors.
= 24×3²×5×7²

= 16×9×5×49

= 35280.

For Example- Find the H.C.F. of 72, 126 and 270.
Explanation: Using Prime factorisation method

72 = 2×2×2×3×3 = 2³×3²

126 = 2×3×3×7 = 21×3²×71

270 = 2×3×3×3×5 = 21×3³×51

H.C.F. of the given numbers = the product of common factors with least index
= 21×3²

= 2×3×3 = 18