Question

Find the number of two-digit prime numbers which remain prime even on inverting the position of its digits.

Answer 1

Hint: We have this simple question. There is no big concept or formula implementing in this particular question. The approach is very simple if we know the elementary idea of prime numbers. We know that prime numbers are the natural numbers greater than 1 that have only 2 factors i.e. 1 and themselves.
In other words, we can say that "a natural number greater than 1 which is divisible by 1 and itself only is a prime number".
For example, we have 2, 3, 5, 7, and so on.

Complete step-by-step solution:
Now let’s come to our question,
First thing, we have to find all 2 digit prime numbers.
So, here we have, 2, 3, 5, 7 as they are single digit, therefore discard them.
Now, further we have:
\[11,13,17,19,23,29,31,37,41,43,47,53,59,61,67,71,73,79,83,89,97\]
Now, after reversing the digits we have:
11 - Prime number, therefore accept this (Accepted)
31 – Prime number (Accepted)
71 – Prime number (Accepted)
91 - It is not a prime number as $\left( 13\times 7=91 \right)$ (Rejected)
32 - Even number (Rejected)
92 - Even number (Rejected)
13 - Prime number (Accepted)
73 - Prime number (Accepted)
14 - Even number (Rejected)
34 - Even number (Rejected)
74 - Even number (Rejected)
35 - Not prime as 35 is divisible by 5 also $\left( 5\times 7=35 \right)$ (Rejected)
95 - Not prime as $\left( 5\times 19=95 \right)$ (Rejected)
16 - Even number (Rejected)
76 - Even number (Rejected)
17 - Prime number (Accepted)
37 - Prime number (Accepted)
97 - Prime number (Accepted)
38 - Even number (Rejected)
98 - Even number (Rejected)
79 - Prime number (Accepted)
Hence, from above evaluation, we found that total 9 two-digit prime numbers are there.
Therefore, \[11,31,71,13,73,17,37,97,79\] is the answer.

Note: Sometimes students think that 11 is not yielding any different number on inversion. So, they reject the prime number 11. But, this is not the case, we don't have to assume anything until or unless it is mentioned in the question. Here, in the question, there is no information that "on inverting the digits, you should get different number". It may be same like in case of 11. Hence, our solution would be 9 (total 9 two-digit prime numbers following given criteria).