Question

A prime number is called a super-prime if doubling it and then subtracting 1 results in another prime number, then the number of Superprimes less than 15 is _ _ _ _ _ _.
A. $2$
B. $3$
C. $4$
D. $8$

Answer 1

Hint: A prime number is a number which is divisible by 1 and itself only. For example, 5 is divisible by 1 and 5 itself, so 5 is a prime number. Prime number starts from 2 and then 3,5…
Super-prime is a prime number which when doubled and 1 is subtracted, it gives another prime number. For example, prime number 3
After doubling 3 we get
$3 \times 2 = 6$
Then 1 is subtracted from 6
$6 - 1 = 5$
And now 5 is also a prime number. Hence 3 is super-prime.

Complete step by step solution:
We list down all prime numbers which are less than 15.
$2,3,5,7,11\& 13$
We check for 2
On doubling it and subtracting 1 we get
$
  \left( {2 \times 2} \right) - 1 \\
   = 4 - 1 \\
   = 3 \\
$
Here 3 is also prime. Therefore 2 is super-prime.
Similarly, we check for 3
$
  \left( {3 \times 2} \right) - 1 \\
   = 6 - 1 \\
   = 5 \\
$
Here 5 is also prime. Therefore 3 is super-prime.
Similarly, we check for 5
$
  \left( {5 \times 2} \right) - 1 \\
   = 10 - 1 \\
   = 9 \\
$
Here 9 is not prime. Therefore 5 is not super-prime.
Similarly, we check for 7
$
  \left( {7 \times 2} \right) - 1 \\
   = 14 - 1 \\
   = 13 \\
$
Here 13 is also prime. Therefore 7 is super-prime.
Similarly, we check for 11
$
  \left( {11 \times 2} \right) - 1 \\
   = 22 - 1 \\
   = 21 \\
$
Here 21 is not prime. Therefore 11 is not super-prime.
Similarly, we check for 13
$
  \left( {13 \times 2} \right) - 1 \\
   = 26 - 1 \\
   = 25 \\
$
Here 25 is not prime. Therefore 13 is not super-prime.
Hence, we have 3 super-primes ${\text{2,3& 7}}$.

Hence, Option B is correct.

Note: If ${\text{n}}$ is a prime number and $2n - 1$ is also prime, then ${\text{n}}$ is called super-prime otherwise not.