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Number of Divisors/Factors

If one integer can be divided by another integer an exact number of times, then the first number is said to be a multiple of the second, and the second number is said to be a factor of the first.
 
For example, 48 is a multiple of 6 because 6 goes into 48 an exact number of times (8 times in this case). Similarly, 6 is a factor of 48. On the other hand, 48 is not a multiple of 5, because 5 does not go into 48 an exact number of times. So, 5 is not a factor of 48.
 
When we talk about the number of divisors of any number, we are basically talking about positive integral divisor of that number. Likewise, it can be observed that 20 is having six divisors namely 1, 2, 4, 5, 10 and 20.
 
Its very essential here to understand the mechanism of formation of divisors.
Since 20 = 22 × 51
 
We will take three power of 2 viz., 20, 21 and 22 and two powers of 5 viz., 50 and 51.
 
Divisors will come from all the possible arrangements of powers of 2 and powers of 5.
20 × 50 = 1
20 × 51 = 5
21 × 50 = 2
21 × 51 = 10
22 × 50 = 4
22 × 51 = 20

Now, to find out the number of divisors of any number, we can use the following formula
If N is any number which can be factorised like N = ap × bq × cr ×…, where ab and c are prime numbers.
Number of divisors = (p + 1) (q + 1) (r + 1)…
 
Example-1
Find the number of divisors of N = 420.
Solution
N = 420 = 22 × 31 × 71 × 51
So, the number of divisors =
(2 + 1) (1 + 1) (1 + 1) (1 + 1) = 24
 
 
Example-2
Find the total number of even and prime divisors of N = 420.
Solution
N = 420 = 22 × 31 × 71 × 51
Odd divisors will come only if we take zero power of 2 (since, any number multiplied by any power (≥1) of 2 will give us an even number)
So, odd divisors will come if we take N1 = 20 × 31 × 71 × 51
So, number of odd divisors = (0 + 1) (1 + 1) (1 + 1) (1 + 1) = 8
So, total number of even divisors = total number of divisors – number of odd divisors = 24 – 8 = 16
Alternatively, we can also find out the number of even divisors of N = 420 directly (or, in general for any number).
420 = 22 × 31 × 71 × 51
To obtain the factors of 420 which are even, we will not consider 20, since 20 = 1
So, the number of even divisors of 420 = (2) (1 + 1) 
(1 + 1) (1 + 1) = 16
(We are not adding 1 in the power of 2, since we are not taking 20 here, i.e., we are not taking one power of 2.)
Prime divisor = 4 (namely 2, 3, 5 and 7 only)
 

Example-3
N = 27 × 35 × 56 × 78. How many factors of N are divisible by 50 but not by 100?
Solution
All the factors which are divisible by 50 but not divisible by 100 will have atleast two powers of 5, and one power of 2.
And its format will be 21 × 52+y.
So, number of divisors = 1 × 6 × 5 × 9 = 270
Alternatively, this is equal to (Number of factors divisible by 50) – (Number of factors divisible by 100).
 

Condition for two divisors of any number n to be co-prime to each other

Let us see it for N = 12
 
Total number of factors of 12 = 6 (namely 1, 2, 3, 4, 6, 12)
 
Now, if we have to find out the set of factors of this number which are co-prime to each other, we can start with 1.
 
Number of factors which are co-prime to 1 = 5 (namely 2, 3, 4, 6, 12)
 
Next in line is the number of factors which are co-prime to 2 = 1 (namely 3)
 
Number of factors co-prime to 3 = 1 (namely 4)
 
So, the total number of set of factors for 12 which are co-prime to each other = 6
 
So, we can induce that if we have to find out the set of factors which are co-prime to each other for N ap × bq, will be equal to [(p + 1) (q + 1) – 1 + pq].
 
If there are three prime factors of the number, i.e., N = ap × bq × cr, then set of co-prime factors can be given by [(p +1) (q + 1) (r + 1) – 1 + pq qr pr + 3 pqr]
 
Alternatively, we can find out the set of co-prime factors of this number by pairing it up first and then finding it out with the third factor.
 
Example
Find the set of co-prime factors of the number N = 720.
Solution
720 = 24 × 32 × 51
Using the formula for three prime factors [(p + 1) (q + 1) (r + 1) – 1 + pq + qr + pr + 3 pqr]
 
We get, [(4 + 1) (2 + 1) (1 + 1) – 1 + 4.2 + 2.1 + 4.1 + 3.4.2.1] = 67
 
Alternatively, let us find out first for 24 × 32 = [(4 + 1) (2 + 1) – 1 + 4.2] = 22
 
Now p22 × 51 will give us [(22 + 1) (1 + 1) – 1 + 22.1] = 67
 

Sum of Divisors

Like the number of divisors of any number, we can find out the sum of divisors also.

If N is any number which can be factorised like N = ap × bq × cr ×…, where ab and c are prime numbers. Then sum of the divisors
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