Factors 1) A factor of a number is an exact divisor ... - Educomp Online

CLASS NOTES
CLASS : VI
SUBJECT : MATHEMATICS
TOPIC : PLAYING WITH NUMBERS WEEK : SEPTEMBER 2012
TEACHER’S NAME:
: NANDINI DHIR
Factors 1) A factor of a number is an exact divisor of that number
Example – Factors of 50
1 × 50 = 50
2 × 25 = 50
5 × 10 = 50
Factors of 50 are 1, 2, 5, 10, 25, 50
Factor 2 exactly divides 50
2) 1 is a factor of every number and it is the smallest factor
3) Every number is a factor of itself and number given is the greated factor
Example: - 50 is greatest factor of 50.
4) Every factor is less than or equal to the given number.
5) Number of factors of a given number are finite.
Multiples
1) Number multiplied by 1, 2, 3,………… etc gives its multiples
Example multiples of 6 are 6, 12, 18 ……….…
2) Every multiple of a number is greater than or equal to that number
3) The numbers of multiples of a given numbers is infinite.
4) Every numbers is a multiple of itself
Prime numbers
The number other than 1 whose only factors are 1 and the number itself
Example – 3, 5 are prime numbers

CLASS NOTES
Composite numbers-
Example -: 4, 9
Numbers having more than two factors,
Perfect numbers
A number for which sum of all its factors is equal to twice the number
Example-: 6 is a perfect number
Factors of 6 are 1, 2, 3, 6
1 + 2 + 3 + 6 = 12 = 2 × 6
Sum of all factors = twice the number
Even Numbers-
Multiples of 2 are even numbers. A numbers with 0,2, 4, 6, 8 at ones place is an
even numbers.
Odd numbers
Numbers which are not multiples of 2
Example – 1, 23, 15 are odd number.
Some facts
i) 2 is the smallest even prime number
ii) Every prime number except 2 is odd
iii) 1 is neither prime nor composite
Co- prime numbers
Two numbers with only 1 as a common factor
Example 9 and 16.
Twin primes
Two prime numbers whose difference is 2.
Example 3, 5 are two prime numbers and their difference 5 – 3 = 2
Divisibility tests
Divisibility by 2– If a number has 0, 2, 4, 6, 8 at its ones place it is divisible by 2
Example- 246 is divisible by 2 as it has 6 at its ones place.

CLASS NOTES
Divisibility by 3- If the sum of the digits is divisible by 3 or a multiple of 3, then the
numbers is divisible by 3
Example = 1893
Sum of digits 1 + 8 + 9 + 3 = 21 and 21 ÷ 3 = 7,
As sum of digits of 1893 is divisible by 3, 1893 is divisible by 3
Divisibility by 4 – A number with 3 or more digits is divisible by 4 if the number
formed by its last two digits is divisible by 4
Example 112 is divisible by 4 as 12 is divisible by 4
Divisibility by 5 - A number which has either 0 or 5 in its ones place is divisible by 5
Example 505 is divisible by 5 as it has 5 at ones place
Divisibility by 6
It a number is divisible by 2 and 3 both then it is divisible by 6 also
Divisibility by 8- A number with 4 or more digits is divisible by 8, if the number
formed by the last three digits is divisible by 8.
Example – 1248
1248 is divisible by 8 ( 248 ÷ 8 = 31). Therefore 1248 is divisible by 8
Divisibility by 9 – It the sum of the digits of a number is divisible by 9, then the
number is divisible by 9.
Example 19287, 9 + 1 + 2 + 8 + 7 = 27 since 27 is divisible by 9, 19287 is divisible by
9
Divisibility by 11- If the difference between the sum of digits at odd Places (from the
right) and sum of the digits at even places (from the right) is either 0 or multiple of 11,
then the number is divisible by 11.
Example 54560
Sum of the digits at odd places = 0 + 5 + 5
= 10
Sum of the digits at even place = 6 + 4
= 10
Difference = 10 – 10 = 0
Thus number 54560 is divisible by 11
Some more divisibility rules
Some more divisibility rules

CLASS NOTES
1) If a number is divisible by another number then it is divisible by each of the
factors of that number 24 is divisible by 8, Factors of 8 are 1, 2, 4, 8 24 is
divisible by 1, 2, 4 and 8.
2) If a number is divisible by two co-prime numbers then it is divisible by their
product also
4 and 5 are two co primes which divide 80, product 4 × 5 = 20 also divides
80
3) If two given numbers are divisible by a number, then their sum is also
divisible by that number.
16 and 20 are divisible by 4, then 16 + 20 = 36 is also divisible by 4
then 16 + 20 = 36 is also divisible by 4
4) If two given numbers are divisible by a number, then their difference is also
divisible by that number.
20 and 25 are divisible by 5, then 25 – 20 = 5 is also divisible by 5.
Prime factorization
When all the factors of a given number are prime, the prime factors with product sign
give the prime factorization
36 = 2 × 18
= 2 × 2 × 9
36 = 2 × 2 × 3 × 3
Prime factorization of 36 is 2 × 2 ×3 ×3
Factor tree
36
2 × 18
2 × 2 × 9
Division method
2 × 2 × 3 × 3

CLASS NOTES
2 36
2 18
3 9
3 3
1
Prime factorization of 36 = 2 × 2 × 3 ×3
(HCF)
Highest common factor or Greatest common divisor (GCD)
HCF OR GCD is the highest of all common factors
HCF OF 18, 48
a) Factors of 18 = 1, 2, 3, 6, 9, 18
Factors of 48 = 1, 2, 3, 4, 6, 8, 12, 16, 24, 48
Common factors are 1, 2, 3, 6,
HCF is 6
b) HCF using prime factorization
HCF = 2 × 3 = 6
c) Division method
HCF is 6
Lowest Common Multiple

CLASS NOTES
a) LCM of 4, 6
Multiple of 4 = 4, 8, 12, 16, 20, 24 ………………..
Multiple of 6 = 6, 12, 18, 24, ……………
Common multiple = 12, 24
LCM = 12
b) Using prime factorization
4 = 2 × 2
6 = 2 × 3
LCM = 2 × 2 × 3 = 12
Division method
2 4, 6
2 2, 3
3 1, 3
1, 1
LCM = 2 × 2 × 3 = 12