Question

a natural number n has exactly two divisors and (n+1) has exactly three divisors the number of divisors of (n+2)

1)2. 2)3. 3)4. 4) depends on value of n

Answer 1

Given : a natural number n has exactly two divisors and (n+1) has exactly three divisors

To find : number of divisors of (n+2)

Solution:

(n+1) has exactly three divisors

To have exactly  three divisors number must be Square

=> n + 1  = k²

where k  is Prime  

=> n = k² - 1  

=> n = ( k + 1) ( k - 1)

now its given that  n has exactly two divisors  

=> ( k + 1)  & ( k - 1 ) are also  prime

hence   k - 1 ,  k  &  k + 1  are prime

this is only possible

if k = 2

=> n = 2² - 1 = 3

n = 3  has exactly two divisor  1  & 3

n + 1 = 4  has exactly 3 Divisor   1 , 2  & 4

n + 2 = 5   Has two Divisor  1 & 5

number of divisors of (n+2)  = 2

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