Maths formulas # 05 :)
PROBLEMS ON H.C.F AND L.C.M
- Factors and Multiples:
If number a divided
another number b exactly, we say that a is
a factor of b.
In this case, b is
called a multiple of a.
- 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:
- Factorization
Method: Express
the 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.
- Division
Method: Suppose
we have to find the H.C.F. of two given numbers, divide the larger 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 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 number.
Similarly, the H.C.F. of more than
three numbers may be obtained.
- 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.
There are two methods of finding the
L.C.M. of a given set of numbers:
- Factorization
Method: 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.
- Division Method (short-cut): Arrange the given numbers in a rwo in any order. Divide by a number which divided 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.
- Product of two numbers = Product of their H.C.F. and L.C.M.
- Co-primes: Two numbers are said to be co-primes if their H.C.F. is 1.
- H.C.F. and L.C.M. of Fractions:
1. H.C.F. =
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H.C.F. of Numerators
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L.C.M. of Denominators
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2. L.C.M. =
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L.C.M. of Numerators
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H.C.F. of Denominators
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- H.C.F. and L.C.M. of Decimal Fractions:
In a 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.
- 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.
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