Maths-
General
Easy

Question

Total number of divisors of 480, that are of the form 4n + 2, n greater or equal than 0, is equal to :

  1. 2    
  2. 3    
  3. 4    
  4. None of these    

Hint:

In order to solve this question, we should know that the number of the divisor of any number 

xa to the power of m b to the power of n c to the power of p...... space w h e r e space a comma b comma 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


    In this question, we have been asked to find the total number of divisors of 480 which are of the form 4n + 2,n greater or equal than 0 
    To solve this question, we should know that the total number of divisors of any number x of the form a to the power of m b to the power of n c to the power of p...... space w h e r e space a comma b comma c are prime numbers and is given by (m + 1) (n + 1) (p + 1)….. we know that 480 can be expressed as
    480 space equals space 2 to the power of 5cross times 3 cross times 5

    So, according to the formula, the total number of divisors of 480 are (5 + 1) (1 + 1) (1 + 1) = 6×2×2=24
    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) =  2×2=4, 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, 480 space equals space 2 squared space left parenthesis space 2 to the power of 8 cross times 3 cross times 5 right parenthesis
    So, the number of divisors that are multiples of 4 are (3 + 1) (1 + 1) (1 + 1) = 4×2×2 = 16.
    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, n0
    .
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    .

    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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