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➤ A number can be expressed as a product of its factors. e.g.,
| 1×4=4 2×2=4. so, 1, 2 and 4 are factors of 4. |
1×6=6 2×3=6. so, 1, 2, 3 and 6 are factors of 6. |
1×16= 16 2× 8 = 16 4× 4 = 16 so, 1, 2, 4, 8 and 16 are factors of 16. |
➤ Every number is factor of itself (4, 6, and 16 in above examples).
➤ Every factor is less than or equal to the given number.
➤ Number of factors of a given number are finite.
How to find factors of a given number
↪ Start with one and check if it can be multiplied with any number (less than the given number) to get the given number. Obviously, one is multiplied with the number itself to get the number. So, 1 and the number itself are always be the factors of the number. Write it as a product of factors.↪ Proceed with next whole number & check if it can be multiplied with any number (less than the given number) to get the given number. If yes we will get two more factors. Write it as a product of factors.
↪ Go on until the factors are not repeated.
Example: let's find the factors of 16
1×16= 16, (1 & itself)
2× 8 = 16, (Next number is 2 which is when multiplied with 8 gives 16)
4× 4 = 16, (4 is repeated, need not to go further)
∴ Factors of 16 are → 1, 2, 4, 8 & 16
Perfect number
A number for which sum of all its factors is equal to twice the number is called a perfect number. e.g.,6 (1+2+3+6=12 =2×6),
28 (1+2+4+7+14+28= 56= 2×28)
Common factors
Factors in common between two or more numbers are called common factors. e.g., 1 and 2 are common factors of 4 and 6.Multiples
"Multiple of a number is the product of the number itself with another number."e.g.,
4×1=4
4×2=8
4×3=12
...
So, multiples of 4 are 4, 8, 12...
First five multiples of 7 are 7, 14, 21, 28, 35.
➤ A number is multiple of each of its factors.
➤ Every number is multiple of itself.
➤ Every multiple of a number is greater than or equal to that number.
➤ The number of multiples of a given number is infinite.
Common multiples
Multiples in common of two or more numbers are called common multiples.e.g., 28 is one of common multiples of 4 and 7.Even & Odd numbers
"An even number is a number which is a multiple of two."➤ It is divisible by 2. e.g., 2, 4, 6, ...
➤ 2 is the first or smallest even number.
"An odd number is a number which is not a multiple of two."
➤ It is not divisible by 2. e.g., 1, 3, 5, ...
➤ 1 is the first or smallest odd number.
Prime and Composite numbers
"The numbers other than 1 whose only factors are 1 and the number itself are called Prime numbers."| e.g., | 2 | 1×2=2 |
| 3 | 1×3= 3 | |
| 5 | 1×5= 5 | |
| ... |
➤ Every prime number except 2 is odd.
Sieve of Eratosthenes
Method to find prime numbers between 1 & 100- devised by Greek Mathematician Eratosthenes in 3rd century B.C.Co-primes / Relative primes / Mutually primes
Two numbers having only 1 as a common factor are called co-primes or relative primes or mutually primes. In other words, two numbers whose HCF/GCD is 1 are called co- primes. e.g., 3 and 5, 3 and 8, 5 and 9 etc.Twin Primes
Two prime numbers whose difference is two are called twin primes. 3 and 5, 5 and 7, 11 and 13 etc.Composite Numbers
"Numbers having more than two factors are called Composite numbers."e.g.,
| 4 |
1×4= 4 2×2= 4 |
| 6 |
1×6= 6 2×3= 6 |
| 8 |
1×8= 8 2×4= 8 |
0 is neither prime not composite.
It has infinite number of factors (any nonzero whole number divides zero, so it is not prime.It cannot be written as a product of two factors, neither of which is itself, so one is also not composite.
[It falls in a class of numbers called zero-divisors. These are numbers such that, when multiplied by some nonzero number, the product is zero].
1 is neither prime nor composite.
It has only one factor, 1 itself, so it is not prime.It cannot be written as a product of two factors, neither of which is itself, so zero is also not composite.
[It falls in a class of numbers called units. These are the numbers whose reciprocals are also whole numbers].
Factorisation
"When the number is expressed as a product of its factors, then the number is said to be factorised."
e.g.,
| 24=2×12 | or, 24=4×6 | or, 24=3×8 |
Prime Factorisation
"When the number is factorised into its prime factors, then the factorisation is said to be prime factorisation."
e.g.,
| 24=2×12 =2×2×6 =2×2×2×3 |
24=4×6 =2×2×6 =2×2×2×3 |
24=3×8 =3×2×4 =3×2×2×2 =2×2×2×3 |
∴Prime factorisation of 24 = 2×2×2×3, every factor must be a prime factor.
➤ There is only one (unique!) set of prime factors for any number.
Factor tree method
Repeated division method
Start dividing the number with smallest possible prime number, if it become indivisible use another least possible prime number to divide the number left, continue till the number become 1.
Time to Think
1) Which is smallest composite number?2) Which is smallest prime number?
3) Are co-prime numbers are also prime numbers?
4) Can a number have a factor greater than itself?
5) A number can have infinite number of multiples-true or false?
Answer
1) 4.2) 2.
3) Not necessarily, two composite numbers can be co-primes.
4) No.
5) True.
| Whole Numbers | |