HCF & LCM (Telugu?)
Pre-requisite: Number System, Prime Numbers, Prime Factorization, Multiplication,
Number Comparison (Lowest, Largest, Equal etc)
Target Audience: Standard V to Standard VII.
In Syllabus – Standard V.
Revision – Standard VI & VII.
Factors (Telugu?) of a number:
Factors of a number are all the number less than or equal to the number, which divide the
number (leaves remainder 0).
For example, factors of 8 are: 1, 2, 4, and 8.
Rules to find out factors of a number:
a) „1‟ is a factor of all the numbers.
b) The prime factors of a number are factors of that number.
c) The products of numbers in the prime factorization, taken two at time, three at a time
and so forth, until the product is equal to the number, are factors of that number.
d) The number is a factor of itself.
Example, find out factors of 36.
a) By rule „a‟ above 1 is a factor of „36‟
b) Prime factors of 36:
Prime factorization of 36 = 2 * 2 * 3 * 3
Prime factors of „36‟ are 2 & 3. According to rule „b‟ above „2‟ and „3‟ are factors of 36.
c) Prime factorization of „36‟: 2 * 2 * 3 * 3
Taking 2 numbers at a time from the numbers in prime factorization:
2*2=4
2*3=6
3*3=9
All the possible combinations are taken, no more products possible taking 2 numbers at a
time.
Taking 3 numbers at a time from the numbers in prime factorization:
2 * 2 * 3 = 12
2 * 3 * 3 = 18
No more combinations possible. Move to 4 numbers at a time now.
Taking 4 numbers at a time from the numbers in prime factorization:
2 * 2 * 3 * 3 = 36
Since product is equal to the number (36), stop here.
d) The number „36‟ is a factor of itself.
Combining a, b, c, d:
The factors of 36 are: 1, 2, 3, 4, 6, 9, 12, 18, and 36
Composite Factors (Telugu?):
Composite factors are all factors except „1‟ and prime factors.
Composite factors of „36‟ are: 4, 6, 9, 12, 18, and 36
(After removing „1‟ and prime factors (2 and 3) from the factors of „36‟)
Common factors of two or more numbers:
The common factors of two or more numbers are the numbers common in the factors of the
numbers.
Rule to find out common factors:
a) Find out factors of the numbers
b) List down common numbers in the factors of the numbers
Find out common factors of „36‟ and „48‟:
Factors of „36‟ are: 1, 2, 3, 4, 6, 9, 12, 18, and 36
Factors of „48‟ are: 1, 2, 3, 4, 6, 8, 12, 16, 24, and 48
** Find out factors of „48‟ using the method illustrated above.
Common factors are: 1, 2, 3, 4, 6, and 12
Highest Common Factor (HCF):
HCF is highest among the common factors of two or more numbers.
HCF of „36‟ and „48‟ is 12, because 12 is highest among common factors {1, 2, 3, 4, 6, and 12}
HCF is also called Greatest Common Divisor (GCD) or Greatest Common Factor (GCF)
Least Common Multiple (LCM)
Multiples of a number can be obtained by multiplying the number with the natural numbers
{1,2,3,4….}
Multiples of 3: 3*1, 3*2, 3*3, 3*4, 3*5, 3*6, 3*7……
3, 6, 9, 12, 15, 18, 21…..
Common multiple of two or more numbers:
All the multiples of the give numbers which are common are called common multiples of the
numbers.
For example:
Multiple of 2: 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24….
Multiple of 3: 3, 6, 9, 12, 15, 18, 21, 24……
So the multiples which are common are: 6, 12, 18, 24…..
*** Note that there could be more common multiples. The above multiples are common
in the list of multiples we have generated.
Least common multiple (LCM) of two or more numbers is the number with lowest numeric
value among common multiples. In the above example, in the common multiples of 2 and 3, „6‟
has the lowest numeric value. Hence LCM of 2 and 3 is 6.
*** The above description of LCM is to help students understand the meaning of LCM. The
objective is to make them understand that LCM is “Lowest Number” among “Common
Multiples”.
Finding LCM by Prime Factorization Method:
The LCM of two or more numbers is calculated by multiplying, all the prime factors of numbers
occurring the highest number of times as it occurs in the prime factorization of any of the given
numbers.
Let us find out LCM of 8, 12 and 20:
Prime Factorization of 8: 2 * 2 * 2
Prime Factorization of 12: 2 * 2 * 3
Prime Factorization of 20: 2 * 2 * 5
Step 1: List out all the distinct prime numbers occurring in the prime factorization of numbers.
Step 2: Take one prime number at a time from the list obtained from Step 1, and find out where it
is used highest number of times. Select those many numbers of the prime numbers for
calculation of LCM
Step 3: Multiply all the numbers obtained in step 2 to get LCM.
1) Distinct prime numbers in the prime factorization of 8, 12 and 20 are {2, 3, 5}
2) 2 has occurred highest thrice in prime factorization of 8: 2 * 2 * 2
3 has occurred highest once in prime factorization of 12: 3
5 has occurred highest once in prime factorization of 20: 5
3) Multiply (2 * 2 * 2) * 3 * 5 = 8 * 3 * 5 = 120
So LCM of 8, 12 and 20 is 120.
Rule of Divisibility:
1) A number ending with {0, 2, 4, 6, 8} is always divisible by 2. For example 20, 12, 4, 16,
28e
2) If the sum of the digits of a number is divisible by 3, the number is divisible by 3.
For example – 27, sum of digits = 2 + 7 = 9 is divisible by 3, so 27 is divisible by 3
129, 1 + 2 + 9 = 12 is divisible by 3, hence 129 is also divisible by 3.
3) A number ending with {0, 5} is always divisible by 5. For example 10, 25, 50, 15
LCM by Division method:
1) Use the “Rule of Divisibility” to find out smallest prime number dividing all the
numbers. Check the divisibility with 7 and 11 separately as the rule of divisibility does
not have a quick method to check divisibility with 7 and 11.
2) Divide the numbers by the smallest prime number thus found in step 1 and write down
the quotients.
3) Repeat steps 1 through 3 until there is no prime dividing all the numbers.
Find out LCM of 12, 16, and 20:
2
2
12, 16, 20
6,
8, 10
3,
4, 5
1) The smallest prime number dividing 12, 16 and 20 is 2. Quotients are 6, 8 and 10
2) The smallest prime number dividing 6, 8 and 10 is 2. Quotients are 3, 4 and 5
3) There is no common prime dividing 3, 4 and 5. So stop here. LCM of 12, 16 and 20 is
2 * 2 * 3 * 4 * 5 = 240
LCM is written as [12, 16, 20] = 240