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
Answers
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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