Aptitude Questions
HCF and LCM
Factor: A factor of a given number is every number that
divides exactly into that number
Example : Factors of 12 are
1,2,3,4,6,12
Prime factorization
If a natural number is expressed as the
product of prime numbers, then the factorization of the number is called its
prime (or complete) factorization.
A prime factorization of a natural number
can be expressed in the exponential form.
For example:
(i) 90 = 2×3×3×5 = 2 × 2³×3
(ii) 420 = 2×2×3×5×7 = 2^{3} ×3^{2}×5×7.
Common Factor: A number which exactly divides all
the given numbers is called common factor of those numbers.
Example Find all the common factors of
18 and 24.
Factors of 18 are 1, 2, 3, 6, 9,18.
Factors of 24 are 1, 2, 3, 4, 6, 8,12, 24
The common factors of 18 and 24 are 1, 2, 3 and 6
Highest Common Factor: The greatest number that exactly
divides all the given numbers is called Highest Common Factor (HCF). It is also
called Greatest Common Divisor (GCD) or Highest Common Divisor (HCD )or
Greatest Common Factor ( GCF)
To find HCF of Numbers:
Method 1:
Prime Factorization method: This method deals with the prime
factors of numbers. It involves the following steps
Step1 : Express each one of the given
numbers as the product of prime factors.
Step2 : Identify common prime factors
Step3 : Compute the product of all common
prime factors, provided each common prime factor must appear in the prime
factorization of all the given numbers.
The product of least powers of common
prime factors gives HCF.
Example : Find the HCF
of 18 ,12,24 .
Using Prime factorization method to Find HCF
18=2 x 3 X 3 => 18 = 2 x 3^{2}
12=2 x 2 x 3=> 24 =2^{2}x 3
24=2 x 2 X 2 X 3 => 2^{3} x 3
H.C.F. of the given numbers = the product of common factors with
least index
Then HCF= 2 x 3 =6
Find the H.C.F. of 144,270 and 630
.Solution
Using Prime
factorization method
72 = 2×2×2×3×3 = 2³×3²
270 = 2×3×3×3×5 = 2^{1}×3³×5^{1}
72 = 2×2×2×3×3 = 2³×3²
270 = 2×3×3×3×5 = 2^{1}×3³×5^{1}
630 =
2×3×3×5× 7 = 2^{1}×3²×5×7^{1}
H.C.F. of the given numbers = the product of common factors with least index
HCF = 2^{1}×3² = 2×3×3 = 18
H.C.F. of the given numbers = the product of common factors with least index
HCF = 2^{1}×3² = 2×3×3 = 18
Method 2
Division Method to Find HCF of two numbers
This
method involves the following steps
Step 1 : Divide the larger number by the smaller one to obtain the
remainder
Step 2 : If remainder is 0, the divisor is the required HCF
Step 3: If remainder is not zero, take this
remainder as a divisor and the first divisor as the dividend
Step 4 : Repeat the process till zero is obtained as a remainder.
The
last divisor is the required HCF
Example: HCF OF 84,140

Division Method to Find HCF of more than two
numbers is find HCF of any two numbers as H1, and
find HCF of H1 and the numbers and so on. The last HCF obtained is the HCF of
all the given numbers.
Multiple: When we multiply a given whole
number by any other whole number, the result is a multiple of that number.
For example, 5 is the first multiple of 5 (because 5 x 1 = 5), 10 is the
second multiple of 5, and so on.
Example : Multiples of 12 =
12,24,36,48,60,72,..
Common Multiple: A number which is exactly divisible
by all the given numbers is common multiple.
Example: Common multiple of 3, 5, 6, 8 is
120.
Least Common Multiple (LCM): Definition of LCM The least number which is exactly
divisible by all the given number is LCM.
In other words , LCM of two or more numbers is the smallest number which
is a multiple of each of the number.
LCM is also known
as lowest common dividend ( LCD)
To
Find LCM of Given Numbers:
Method 1:
Prime
Factorization Method To Find the LCM:
Step 1 : Express each number
as a product of prime powers
Step 2 : For each prime factor choose the prime power with largest exponent
among all numbers.
The
LCM is the product of the chosen prime powers
Example : Find the LCM of 12, 15, and 20
12= 2 x 2 x 3=3 x 2^{2 }
15=3 x 5=3 x 5
20=2 x 2 x 5=2^{2 }x 5
Here the prime factors appear in the given
numbers are 2, 3, 5.
Their highest power are 2^{2}, 3
and 5.
LCM=2^{2}´3´5=60
Method II:
Division
Method To Find LCM : Write
all the given numbers in a now divide them by any one of prime numbers. 2, 3,
5, 7, 11 etc. which will divide at least two given numbers? Write down the
quotients and the other undivided numbers in a row below the first. Repeat this
process till you get a row of numbers, which are prime to one another.
The product of all the divisors and the numbers in the last row is their LCM.
Ex: 12, 15, 20,25
312 15 20
, 25
24
5 20 , 25
22
5 10, 25
51
5 5, 25
1 1
1 , 5
LCM=2 x 2 x 3 x 5 x 5 = 3000
LCM of fractions :
LCM of fractions =
LCM of Numerators
HCF of Denominators
Find the LCM of
2
3
,
5
6
,
7
9
LCM of
fractions =
LCM of Numerators
HCF of Denominators
=
LCM of
2,5,7
HCF of 3,6,9
=
70
3
HCF of fractions :
HCF of
fractions =
HCF of Numerators
LCM of Denominators
Find the HCF of
2
3
,
5
6
,
7
9
Explanation
: HCF of fractions =
HCF of Numerators
LCM of Denominators
=
HCF of 2,5,7
LCM of 3,6,9
=
1
18
Some Important Points :
1. HCF of any two
consecutive numbers is always 1
2. If all the given numbers
are prime numbers, then their HCF is 1
3.HCF of relatively prime
numbers or co primes is always 1
4.LCM of any two consecutive
numbers is always equal to their product
5.If all the given numbers
are prime numbers, then their LCM is equal to product of all given prime
numbers
6. LCM of co primes or
relatively prime numbers is equal to their product
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