Finding the Two-Digit Number

This problem involves finding a two-digit number based on two conditions related to its digits and subtraction.

Setting Up the Algebraic Equations

Let the two-digit number be represented as $10x + y$, where $x$ is the tens digit and $y$ is the units digit.

• Condition 1: Sum of digits is 10. This translates to the equation: $x + y = 10 \quad (1)$
• Condition 2: Subtracting 18 results in equal digits. The number after subtraction is $(10x + y) - 18$. The problem states the digits of this *resulting* number are equal. A number with equal digits can be written as $11a$, where $a$ is the repeated digit. So, the equation is: $(10x + y) - 18 = 11a \quad (2)$ Where $a$ is a digit (0-9).

Solving the Number Equations

We can solve this system algebraically.

1. From equation (1), express $y$ in terms of $x$: $y = 10 - x$
2. Substitute this expression for $y$ into the original number ($10x + y$): $10x + (10 - x) = 9x + 10$ So the number is $9x + 10$.
3. Now substitute $9x + 10$ for $(10x + y)$ in equation (2): $(9x + 10) - 18 = 11a$ $9x - 8 = 11a$
4. We need to find a value for the tens digit $x$ (where $x$ is an integer from 1 to 9) such that $9x - 8$ is a multiple of 11. Let's test values for $x$:
   • If $x=1$, $9(1) - 8 = 1$ (Not divisible by 11)
   • If $x=2$, $9(2) - 8 = 10$ (Not divisible by 11)
   • If $x=3$, $9(3) - 8 = 19$ (Not divisible by 11)
   • If $x=4$, $9(4) - 8 = 28$ (Not divisible by 11)
   • If $x=5$, $9(5) - 8 = 37$ (Not divisible by 11)
   • If $x=6$, $9(6) - 8 = 46$ (Not divisible by 11)
   • If $x=7$, $9(7) - 8 = 63 - 8 = 55$. Since $55 = 11 \times 5$, this is a multiple of 11. So $a=5$. This is a valid solution for $x$.
   • If $x=8$, $9(8) - 8 = 64$ (Not divisible by 11)
   • If $x=9$, $9(9) - 8 = 73$ (Not divisible by 11)
   The only valid tens digit is $x=7$.

Determining the Original Number

Using $x=7$ and equation (1) ($x+y=10$):

The tens digit is 7 and the units digit is 3. Therefore, the number is $10(7) + 3 = 73$.

Verification

Let's check if the number 73 satisfies both conditions:

• Condition 1: Sum of digits. $7 + 3 = 10$. (Satisfied)
• Condition 2: Subtracting 18. $73 - 18 = 55$. The digits of the resulting number (55) are equal (5 and 5). (Satisfied)

Both conditions are met. The number is 73.