To identify the required two-digit number, we must satisfy the following criteria: the sum of its digits is 10, and subtracting 18 from it yields a number with identical digits.
Let the two-digit number be denoted as \(10a + b\), where \(a\) represents the tens digit and \(b\) represents the units digit. The given conditions translate to:
1. \(a + b = 10\)
2. \(10a + b - 18\) produces a number with repeating digits, represented as \(11c\). Consequently, \(10a + b - 18 = 11c\).
Rearranging the second equation yields:
\(10a + b = 11c + 18\).
Given that \(11c\) is a two-digit number with identical digits, \(c\) can range from 1 to 9. We begin by solving for \(c\).
Substituting the first condition, \(a + b = 10\), into the derived equation, we get:
\(b = 10 - a\)
Substituting \(b\) into \(10a + b = 11c + 18\):
\(10a + (10 - a) = 11c + 18\)
\(9a + 10 = 11c + 18\)
\(9a = 11c + 8\)
By testing possible values for \(c\), we seek integer solutions for \(a\) and \(b\):
For \(c = 5\), \((11 \cdot 5) + 8 = 55 + 8 = 63\). Therefore:
\(9a = 63\)
\(a = 7\)
Then \(b = 10 - a = 10 - 7 = 3\).
The resulting number is \(10a + b = 10\cdot7 + 3 = 73\).
Thus, the correct number is 73.