Question

Let P be a prime number such that the number P² + 11 has exactly 6 positive divisors. Find P.​

Answer 1

P may be 3

Step-by-step explanation:

As 9+11=20 has 6 positive divisors:1,2,4,5,10 and 20

Answer 2

Given : 'p' be a prime number such that the number p²+11 has exactly 6 positive divisors

To Find : value of p

Solution:

p = 2 => p² + 11  = 15     ahs 4 divisors ( 1 , 3 , 5 , 15)

p =3 => p² + 11  = 20    has  6 exactly  divisors ( 1 , 2 , 4 , 5 , 10 , 20 )

Hence p = 3

after that all prime can be written in form :

p = 6q - 1 , 6q + 1

p² + 11  = (6q - 1)²  + 11  = 36q²  + 1  - 12q  + 11   = 36q² - 12q  + 12

= 12( 3q² - q  + 1)

12 it self has 6 divisors Hence more than 6 divisors

p² + 11  = (6q+ 1)²  + 11  = 36q²  + 1  + 12q  + 11   = 36q² +  12q  + 12

= 12( 3q² + q  + 1)

12 it self has 6 divisors Hence more than 6 divisors

so only possible p  = 3

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