The integer 113 is a prime, and its reverse 311 is also a prime. How many twodigit
primes are there between 10 and 99 which have the same property?
since primes must end in 1,3,7,9 there are only a few decades to check.
13 and 31 , 17 and 71 , 19 and 91 , 37 and 73 , 79 and 97 .
To find the number of two-digit primes with the same property as 113, we need to find primes between 10 and 99 that are also prime when their digits are reversed.
Let's start by listing all the two-digit primes between 10 and 99:
11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97
Now, we need to check which of these primes are also prime when their digits are reversed:
11 - Reversed: 11 (also prime)
13 - Reversed: 31 (also prime)
17 - Reversed: 71 (also prime)
19 - Reversed: 91 (not prime)
23 - Reversed: 32 (not prime)
29 - Reversed: 92 (not prime)
31 - Reversed: 13 (also prime)
37 - Reversed: 73 (also prime)
41 - Reversed: 14 (not prime)
43 - Reversed: 34 (not prime)
47 - Reversed: 74 (not prime)
53 - Reversed: 35 (not prime)
59 - Reversed: 95 (not prime)
61 - Reversed: 16 (not prime)
67 - Reversed: 76 (not prime)
71 - Reversed: 17 (also prime)
73 - Reversed: 37 (also prime)
79 - Reversed: 97 (also prime)
83 - Reversed: 38 (not prime)
89 - Reversed: 98 (not prime)
97 - Reversed: 79 (also prime)
From the list, we can see that there are 8 two-digit primes (11, 13, 17, 31, 37, 71, 73, 79) between 10 and 99 that are prime when their digits are reversed.
To find the number of two-digit primes that satisfy the given property, we can iterate over all the two-digit numbers between 10 and 99, check if each number is a prime, and then check if its reverse is also a prime.
Here is the step-by-step process to find the number of such primes:
1. Initialize a count variable to 0. This count will keep track of the number of two-digit primes that satisfy the given property.
2. Start a loop from 10 to 99 (inclusive) to iterate over all the two-digit numbers.
3. For each number, check if it is a prime. To check if a number is prime, we can iterate from 2 to the square root of that number and divide it by each number. If the number is divisible by any of the numbers in this range, it is not a prime. If it is not divisible by any number, it is a prime.
4. If the number is prime, calculate its reverse by converting the number to a string, reversing the string, and converting it back to an integer.
5. Check if the reverse number is also prime using the same method as in step 3.
6. If both the original number and its reverse are prime, increment the count variable by 1.
7. After the loop ends, the count variable will contain the total number of two-digit primes that have the same property.
Here is a Python code implementation for finding the count:
```python
import math
count = 0
for num in range(10, 100):
if all(num % i != 0 for i in range(2, int(math.sqrt(num)) + 1)):
reverse_num = int(str(num)[::-1])
if all(reverse_num % i != 0 for i in range(2, int(math.sqrt(reverse_num)) + 1)):
count += 1
print(count)
```
Running this code will give you the count of two-digit primes that have the same property.