To solve the problem, we need to find two-digit prime numbers such that when we add 18 to them, the result is another prime number whose digits are reversed. Let's break down the steps to find the solution.
### Step 1: Identify Two-Digit Prime Numbers
First, we need to list all two-digit prime numbers. The two-digit prime numbers are:
11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97.
### Step 2: Set Up the Equation
Let the two-digit prime number be represented as \( p \). According to the problem, if we add 18 to \( p \), we get another prime number whose digits are reversed. If \( p \) has digits \( x \) and \( y \), we can express \( p \) as:
\[ p = 10x + y \]
The reversed number will then be:
\[ r = 10y + x \]
According to the problem, we have:
\[ p + 18 = r \]
Substituting the expressions for \( p \) and \( r \):
\[ 10x + y + 18 = 10y + x \]
### Step 3: Rearranging the Equation
Rearranging the equation gives us:
\[ 10x + y + 18 - x - 10y = 0 \]
This simplifies to:
\[ 9x - 9y + 18 = 0 \]
Dividing the entire equation by 9, we get:
\[ x - y + 2 = 0 \]
Thus, we can express this as:
\[ x - y = -2 \]
or
\[ x = y - 2 \]
### Step 4: Finding Valid Digits
Since \( x \) and \( y \) are the digits of a two-digit number, \( x \) (the tens digit) must be between 1 and 9, and \( y \) (the units digit) must be between 0 and 9. From \( x = y - 2 \), we can deduce:
- \( y \) must be at least 2 (so that \( x \) is at least 0).
- The maximum value for \( y \) is 9, which means \( x \) can be at most 7.
Thus, the valid pairs \((x, y)\) are:
- If \( y = 2 \), then \( x = 0 \) (not valid as \( x \) must be non-zero)
- If \( y = 3 \), then \( x = 1 \) → \( 13 \)
- If \( y = 4 \), then \( x = 2 \) → \( 24 \) (not prime)
- If \( y = 5 \), then \( x = 3 \) → \( 35 \) (not prime)
- If \( y = 6 \), then \( x = 4 \) → \( 46 \) (not prime)
- If \( y = 7 \), then \( x = 5 \) → \( 57 \) (not prime)
- If \( y = 8 \), then \( x = 6 \) → \( 68 \) (not prime)
- If \( y = 9 \), then \( x = 7 \) → \( 79 \)
### Step 5: Check Valid Combinations
Now we check the valid combinations:
1. \( 13 + 18 = 31 \) (both 13 and 31 are prime)
2. \( 79 + 18 = 97 \) (both 79 and 97 are prime)
### Conclusion
The two-digit prime numbers that satisfy the condition are 13 and 79. Therefore, there are **2 such numbers** possible.
