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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