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
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
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