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
x equals a to the power of m space end exponent b to the power of n c to the power of p.... space space 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, n greater or equal than 0 
    In order to solve this question, we should know that the number of the divisor of any number
    x equals a to the power of m space end exponent b to the power of n c to the power of p.... space space 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 480 space equals space 2 to the power of 5.3.5
    S o comma space a c c o r d i n g space t o space t h e space f o r m u l a comma space t h e space t o t a l space n u m b e r space o f space d i v i s o r s space o f space 480 space a r e
space left parenthesis 5 space plus space 1 right parenthesis space left parenthesis 1 space plus space 1 right parenthesis space left parenthesis 1 space plus space 1 right parenthesis space equals space 6 cross times 2 cross times 2 equals 24 space.
    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. 
    S o comma space t h e space t o t a l space n u m b e r space o f space o d d space d i v i s o r s space t h a t space a r e space p o s s i b l e space a r e space left parenthesis 1 space plus space 1 right parenthesis space left parenthesis 1 space plus space 1 right parenthesis space equals space 2 cross times 2 equals 4 space comma space a c c o r d i n g space t o space t h e space p r o p e r t y.
    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 to the power of 5.3.5 
    space S o comma space t h e space n u m b e r space o f space d i v i s o r s space t h a t space a r e space m u l t i p l e s space o f space 4 space a r e space left parenthesis 3 space plus space 1 right parenthesis space left parenthesis 1 space plus space 1 right parenthesis space left parenthesis 1 space plus space 1 right parenthesis space equals space 4 cross times 2 cross times 2 space space equals space 16
    Hence, we can say that there are 16 divisors of 480 which are multiple of 4.
    S o comma space t h e space t o t a l space n u m b e r space o f space d i v i s o r s space w h i c h space a r e space e v e n space b u t space n o t space d i v i s i b l e space b y space 2 space c a n space b e space g i v e n space b y space 20 space – space 16 space equals space 4.
    Thus, total number of divisors of 480, that are of the form 4n + 2, n greater or equal than 0, is equal to 4.

    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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