Question
Total number of divisors of 480, that are of the form 4n + 2, n 
0, is equal to :
- 2
- 3
- 4
- None of these
Hint:
In order to solve this question, we should know that the number of the divisor of any number 
where a, b, c are prime numbers and is given by (m + 1) (n + 1) (p + 1)….. By using this property we can find the solution of this question.
The correct answer is: 4
Detailed Solution
In this question, we have been asked to find the total number of divisors of 480 which are of the form 4n + 2, 
o solve this question, we should know that the total number of divisors of any number x of the form 
where a, b, c … are prime numbers and is given by (m + 1) (n + 1) (p + 1)….. we know that 480 can be expressed as
So, according to the formula, the total number of divisors of 480 are (5 + 1) (1 + 1) (1 + 1) = 
Now, we have been asked to find the number of divisors which are of the form 4n + 2 = 2 (2n + 1), which means odd divisors cannot be a part of the solution. So, the total number of odd divisors that are possible are (1 + 1) (1 + 1) 
according to the property.
Now, we can say the total number of even divisors are = all divisors – odd divisor
= 24 – 4
= 20
Now, we have been given that the divisor should be of 4n + 2, which means they should not be a multiple of 4 but multiple of 2. For that, we will subtract the multiple of 4 which are divisor of 480 from the even divisors.
And, we know that,
So, the number of divisors that are multiples of 4 are (3 + 1) (1 + 1) (1 + 1) 
Hence, we can say that there are 16 divisors of 480 which are multiple of 4.
So, the total number of divisors which are even but not divisible by 2 can be given by 20 – 16 = 4.
Hence, we can say that there are 4 divisors of 480 that are of 4n + 2 form, 
We can also solve this question by writing 4n + 2 = 2(2n + 1) where 2n + 1 is always an odd number. So, when all odd divisors will be multiplied by 2, we will get the divisors that we require. Hence, we can say a number of divisors of 4n + 2 form is the same as the number of odd divisors for 480.
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