H.C.F. AND L.C.M. OF NUMBERS
I. Factors and Multiples :
If a number a divides another number b
exactly, we say that a is a factor of b. In this case, b is called a multiple
of a.
II. Highest
Common Factor (H.C.F.) or Greatest Common Measure (G.C.M.) or Greatest Common
Divisor (G.C.D.):
The H.C.F.
of two or more than two numbers is the greatest number that divides each of
them exactly.
There are two methods of finding the
H.C.F. of a given set of numbers :
1. Factorization Method : Express each one of the given numbers as the
product of prime factors.The product of least powers of common prime factors
gives H.C.F.
2. Division Method: Suppose we have to find the H.C.F. of two
given numbers. Divide the larger number
by the smaller one. Now, 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 the required H.C.F.
Finding
the H.C.F. of more than two numbers : Suppose we have to find the H.C.F. of three numbers. Then, H.C.F. of
[(H.C.F. of any two) and (the third number)] gives the H.C.F. of three given
numbers.
Similarly, the
H.C.F. of more than three numbers may be obtained.
III. Least Common Multiple (L.C.M.) :
The least number which is exactly divisible
by each one of the given numbers is called their L.C.M.
1. Factorization Method of Finding L.C.M.: Resolve each one of the given numbers into a
product of prime factors. Then, L.C.M. is the product of highest powers of all
the factors,
2. Common Division Method {Short-cut Method) of
Finding L.C.M.: Arrange the
given numbers in a row in any order. Divide by a number which divides 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 L.C.M. of the given numbers,
IV. Product of
two numbers =Product of their H.C.F. and L.C.M.
V. Co-primes:
Two numbers are said to be co-primes if
their H.C.F. is 1.
VI. H.C.F. and
L.C.M. of Fractions:
1.H
C F= H.C.F. of Numerators 2.L C M
= L.C.M of Numerators__
L.C.M. of Denominators H.C.F. of Denominators
VII. H.C.F. and
L.C.M. of Decimal Fractions:
In given numbers, make the same number of decimal places by annexing zeros in
some numbers, if necessary. Considering these numbers without decimal point,
find H.C.F. or L.C.M. as the case may be. Now, in the result, mark off as many
decimal places as are there in each of the given numbers.
VIII.
Comparison of Fractions:
Find
the L.C.M. of the denominators of the given fractions. Convert each of the
fractions into an equivalent fraction with L.C.M. as the denominator, by
multiplying both the numerator and denominator by the same number. The
resultant fraction with the greatest numerator is the greatest.
