Question

What is the value of $x$ for which $x,x+1,x+3$ are all prime numbers?

• A. $0$
• B. $1$
• C. $2$
• D. $101$

Answer 1

The correct option is C

$2$

Determine the correct answer

Explanation for the correct options:

Option (C) $x=2$

given expression are $x,x+1,x+3$.

Substitute the value of $x=2$ in the given expression:

$\begin{array}{rcl}x& =& 2\\ & \Rightarrow & x+1=3\\ & \Rightarrow & x+3=5\end{array}$

All the value of expression are prime for $x=2$.

Thus,for $x=2$; $x,x+1,x+3$ are all prime numbers

Hence, option (C) is correct.

Explanation for the incorrect options:

Option (A): $0$

Given expressions are $x,x+1,x+3$.

Substitute the value of $x=0$ in the given expression:

$\begin{array}{rcl}x& =& 0\\ & \Rightarrow & x+1=1\\ & \Rightarrow & x+3=3\end{array}$

All the value of expression $0,1,3$ are not prime for $x=0$

Thus,for $x=0$; $x,x+1,x+3$ are not all prime numbers

Hence, option (A) is incorrect.

Option (B):$1$

Given expression are $x,x+1,x+3$.

Substitute the value of $x=1$ in the given expression:

$\begin{array}{rcl}\mathrm{x}& =& 1\\ & \Rightarrow & \mathrm{x}+1=2\\ & \Rightarrow & \mathrm{x}+3=4\end{array}$

All the values of expression $1,2,4$ are not prime for $x=1$

Hence, option (B) is incorrect.

Option (D):$x=101$

Given expression are $x,x+1,x+3$.

Substitute the value of $x=101$ in the given expression:

$\begin{array}{rcl}x& =& 101\\ & \Rightarrow & x+1=102\\ & \Rightarrow & x+3=104\end{array}$

All the value of expression $101,102,104$ are not prime for $x=101$

Hence, option (D ) is incorrect.

Hence the correct option is Option (D)