How to Find Total Number of Factors for Big Numbers Easily | Number System
In number system the concept of factors of numbers is one of the important sub-topic. In this article, we will discussed about definition of factors of number, formulas for finding number of factors, sum of factors, product of factors, even number of factors, odd number of factors, perfect square factors and perfect cube factors for any number.
Definition of Factors of a number:
Factors of a number any number “ P” refers to all the numbers which are exactly divisible on “P” i.e remainder comes to zero. These factors of numbers are also called divisors of a number.
For example:
Factors of the number 9
- 1 × 9 = 9,
- Also 3 × 3 = 9
So 1, 3, and 9 are factors of 9.
And also -1, -3, and -9 because you get a positive number when you multiply two negatives,
such as (-3)×(-3) = 9
No answer is 1, 3, 9, -1, -3, & -9
But usually factors of numbers consider only positive numbers
Note: fractions of numbers also not consider as a factors
Formulas related to factors of numbers:
- Total number of factors
- Sum of factors
- Product of factors
Take any number “N” and it is to be covert into product of prime numbers (Prime factorization) i.e
N = Ap x Bq x Cr here A, B , C are prime numbers and p,q,and r were respective powers of that prime numbers.
Total numbers of factors for ” N “= (p + 1)(q +1)(r +1)
Sum of all factors of “N”
Product of all factors of “N” = ( N )Total no. of factors/2
Example – 1 : Find the number of factors of 98 and also find the sum and product of all factors
Solution : First write the number 98 into prime factorization
98 = 2 x 49 = 2x 7 x 7
98 = 21 x 72 Here A = 2 , B = 7 , p= 1 , q = 2
Number of factors for the number 98 = (p + 1)(q +1) = 2 x 3 = 6
Sum of all factors of 98 = = 3 x 57 = 171
Product of all factors of number 98 = (98)6/2 = (98)3 = 941192
Example – 2 : Find the number of factors of 588 and also find the sum and product of all factors
Solution : First write the number 588 into prime factorization
588 = 2 x 294 = 2x 2 x 147 = 2 x 2 x 7 x 21 = 2 x 2 x 7 x 7 x 3
588 = 22 x 31 x 72 Here A = 2 , B = 3 , C = 7 , p= 2 , q = 1 and r =2
Number of factors = (p + 1)(q +1)(r +1) = 3 x 2 x 3 = 18
Sum of all factors of 588 = 7 x 4 x 57 = 1596
Product of all factors of number 588 = (588)18/2 = (588)9
Another Concepts in Factors of numbers
- How many factors are odd
- How many factors are even
- Number of perfect square factors
- Number of perfect cube factors
Example – 3 : Find the number of odd, even, perfect square, perfect cube factors of 4500
Solution: First write the number 4500 into prime factorization
4500 = 45 x 100 = 9 x 5 x 10 x 10 = 3 x 3 x 5 x 5 x 2 x 5 x 2
4500 = 22 x 32 x 53 Here consider A = 2 , B = 3 , C = 5 , p= 2 , q = 2 and r = 3
Here identifying that odd number are 3 and 5
Numbers of odd factors of number 4500 = (q + 1 ) (r + 1) = 3 x 4 = 12
Total number of factors = (p + 1)(q +1)(r +1) = 3 x 3 x 4 =36
Numbers of even factors of number = Total number of factors – Numbers of odd factors = 36 – 12 = 24
Number of perfect square factors of number 4500 = 2 x 2 x 2 = 8
( 22 20 , 22 ; 32 30 , 32 & 52 50 , 52 )
Number of perfect cube factors of number 4500 = 1 x 1 x 2 = 2
( 22 20 32 30 , & 52 50 , 53 )
Example – 4 : Find the number of odd, even, perfect square, perfect cube factors of 5040
Solution: First write the number 5040 into prime factorization
5040 = 504 x 10 = 4 x 126 x 5 x 2 = 2 x 2 x 18 x 7 x 5 x 2 = 2 x 2 x 3 x 3 x 2 x 7 x 5 x 2
5040 = 24 x 32 x 71 Here consider A = 2 , B = 3 , C = 7 , p= 4 , q = 2 and r = 1
Here identifying that odd number are 3 and 7
Numbers of odd factors of number 5040 = (q + 1 ) (r + 1) = 3 x 2 = 6
Total number of factors = (p + 1)(q +1)(r +1) = 5 x 3 x 2 = 30
Numbers of even factors of number = Total number of factors – Numbers of odd factors = 30 – 6 = 24
Number of perfect square factors of number 5040 = 3 x 2 x 1 = 6
( 22 20 , 22, 24 ; 32 30 , 32 & 71 70 )
Number of perfect cube factors of number 5040 = 2 x 1 x 1 = 2
( 22 20 , 23 ; 32 30 , & 72 70 )
Related Topics :
Rules for Divisibility of numbers
GCD and LCM Problems & Solutions
Formulas for Sum of n Consecutive numbers
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