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.
$2,3,5,7,11\mathrm{\&}13$
We check for 2
On doubling it and subtracting 1 we get
$$\begin{gathered} \left(2\times 2\right)-1 \\ =4-1 \\ =3 \end{gathered}$$
Here 3 is also prime. Therefore 2 is super-prime.
Similarly, we check for 3
$$\begin{gathered} \left(3\times 2\right)-1 \\ =6-1 \\ =5 \end{gathered}$$
Here 5 is also prime. Therefore 3 is super-prime.
Similarly, we check for 5
$$\begin{gathered} \left(5\times 2\right)-1 \\ =10-1 \\ =9 \end{gathered}$$
Here 9 is not prime. Therefore 5 is not super-prime.
Similarly, we check for 7
$$\begin{gathered} \left(7\times 2\right)-1 \\ =14-1 \\ =13 \end{gathered}$$
Here 13 is also prime. Therefore 7 is super-prime.
Similarly, we check for 11
$$\begin{gathered} \left(11\times 2\right)-1 \\ =22-1 \\ =21 \end{gathered}$$
Here 21 is not prime. Therefore 11 is not super-prime.
Similarly, we check for 13
$$\begin{gathered} \left(13\times 2\right)-1 \\ =26-1 \\ =25 \end{gathered}$$
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.