Aptitude Round Question-H.C.F. & L.C.M. of Numbers-Free Download
Highest Common Factor (H.C.F.) or Greatest Common Measure (G.C.M.) or
Greatest Common Divisor (G.C.D.)
There are two methods of finding the H.C.F. of a given set of numbers:
1. Factorization Method
2. Division method
1. 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.
2. 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:
1. Factorization Method,
2. Division Method (Division Method is short cut method of LCM)
1. 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.
2. Division Method : Arrange the given numbers in a row 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.
. Factors and Multiples
if number a divided another number b exactly, we say that a is a factor of
In this case, b is called a multiple of a.
6. HCF and LCM of Fractions
When solving HCF and LCM questions with fractions these formulas are very
1. H.C.F. = H.C.F. of NumeratorsL.C.M. of Denominators2. L.C.M. = L.C.M. of
NumeratorsH.C.F. of Denominators
Problems on Numbers Formulae
1. (a + b)(a - b) = (a2 - b2)
2. (a + b)2 = (a2 + b2 + 2ab)
3. (a - b)2 = (a2 + b2 - 2ab)
4. (a + b + c)2 = a2 + b2 + c2 + 2(ab + bc + ca)
5. (a3 + b3) = (a + b)(a2 - ab + b2)
6. (a3 - b3) = (a - b)(a2 + ab + b2)
7. (a3 + b3 + c3 - 3abc) = (a + b + c)(a2 + b2 + c2 - ab - bc - ac)
8. When a + b + c = 0, then a3 + b3 + c3 = 3abc.
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