Download 2 - Joy Senior Secondary School

• DEFINITION OF FACTORS AND
MULTIPLES.
• TESTS FOR DIVISIBILITY.
• PRIME AND COMPOSITE NUMBERS
• PRIME FACTORISATION.
• LCM AND HCF.
• RELATION BETWEEN LCM AND HCF.
• FACTORS: A factor is a number that can divide
another number without leaving a remainder.
For eg 24 is divisible by 2,3,4,6,8,12 and 24 so
all these are factors of 24
• MULTIPLES: Multiples of any numbers are the
numbers which are exactly divisible by the
number.
• For eg Multiples of 4 are 4x1=4, 4x2=8, 4x3=12
• DIVISIBILITY RULES FOR 2,3,4,5,6,9,10
• 2
-A number is exactly divisible by 2,if the last digit is 0,2,4,6 or 8. eg 30,
648, 122
• 3
- A number is exactly divisible by 3 if the sum of its digits is exactly
divisible by 3.eg 231 2+3+1=6 and 6/2=3.
• 5
• 6
-A number is exactly divisible by 5 if the last digit is 0 or 5. eg 305, 500
- A number is exactly divisible by 6, if it is exactly divisible by both 2 and
3. eg 48 48/2=24
48/3= 16
• 9
- A number is exactly divisible by 9 if the sum of its digits is exactly
divisible by 9 eg 153 1+5+3=9
and 9/9=1
• 10
- A number is exactly divisible by10 if the last digit is 0.eg 20, 30
• 1 is the smallest factor of every number.
• A factor of any number is always less than
or equal to that number.
• Every number is a multiple of 1.
• Multiples are infinite while factors are
limited.
• Every number is a multiple of itself.
• PRIME NUMBER: A number greater than 1
and having only two factors , is known as prime
number. Eg 2,3,5,7
• NOTE: There is no even prime number except 2.
• COMPOSITE NUMBER: A number greater
than 1 and having more than two factors , is
known as composite number. Eg 4,6,8,10
• TWIN PRIMES: Two prime numbers with a
difference of 2 are called TWIN PRIMES. Eg 3
and 5 (difference=2)
• DEFINITION: When every factor of a number is a prime
number ,the factorisation of that number is known as
prime factorisation of the number .
Example: 18 = 2 x 9
= 2x3x3
2 and 3 are the prime factors
of 18.
Divide to find the prime factors
2
2
2
2
48
24
12
6
3
So prime factorisation of 48 is 2x2x2x2x3
• DEFINITON: The greatest number that
divides the numbers exactly is called the
HCF of the number
• Example: Let us take two composite
numbers 18 and 24
Factors of 18 = 1,2,3,6,9,18
Factors of 24 = 1,2,3,4,6,8,12,24
Common factors = 1,2,3,6
6 is the highest common factor so HCF = 6
• HCF BY PRIME FACTORISATION.
• HCF BY LONG DIVISION METHOD.
• EXAMPLE: Find the HCF of 20 and 24 by prime
factorisation method.
Solution
24
12
6
3
1
20 =2x2x5 24 = 2x2x2x3 common factor =2x2
2 20
2 10
5 5
1
So HCF OF 20 AND 24 IS 4
2
2
2
3
• STEP 1: Divide the greater number by the
smaller number and find the remainder.
• STEP 2: Divide the smaller number or the
divisor by the remainder.
• STEP 3: Continue till you reach the last
divisor. It is the HCF.
• EXAMPLE: Find the HCF of 20 and 24 by
long division method.
1
• SOLUTION:
20 24
20 5
4 20
20
0
Since last divisor is 4. Therefore HCF is 4
• DEFINITION: The least multiple ,which is
common for two or more numbers is their
LCM.
• For example let us take two numbers 2
and 6.
Multiples of 2 – 2,4,6,8,10,12,14,16
Multiples of 6 - 6, 12,18
Common multiples – 6,12 but 6<12
Hence 6 is the LCM of 2 and 6.
METHODS
FOR
LCM
MULTIPLE OR
MEANING
METHOD
PRIME
FACTORISATION
METHOD
COMMON
DIVISION
METHOD
MULTIPLE METHOD – We have already
seen in the definition of LCM.
EXAMPLE: Find the LCM of 8 and 10 by
common division method.
2 8, 10
Solution:
2 4, 5
2 2 ,5
5 1,5
1, 1
So LCM of 8 and 10 = 2 x 2 x 2 x 5 = 40
EXAMPLE :Find the LCM of 12 and
16 by prime factorisation method.
• SOLUTION: LCM of 12 and 16 is
2 12
2 6
3 3
1
2 16
2 8
2 4
2 2
1
12 =2x2x3
Common factors are 2,2, and uncommon
factors are 3,2,2 so multiply 2x2x3x2x2 = 48
16 =2x2x2x2
Therefore LCM = 48
•
•
•
•
PRODUCT OF TWO NUMBERS = LCM X HCF
LCM = PRODUCT OF TWO NUMBERS / HCF
HCF = PRODUCT OF TWO NUMBERS /LCM
EXAMPLE: Find the LCM of two numbers whose product is 120 and
their HCF= 2.
Solution: LCM = PRODUCT OF TWO NUMBERS / HCF
=120/2
=60
so LCM =60
EXAMPLE: Find the HCF of two numbers whose product is 375 and
their LCM =75
Solution: HCF = PRODUCT OF TWO NUMBERS /LCM
= 375/75
=5
So HCF = 5