To solve the equation \( xy \times x = aaa \), we need to clarify the meanings of the variables and what \( aaa \) represents.
Understanding the Equation
- The equation can be rewritten as:
\[
x^2y = aaa
\]
- Here, \( aaa \) represents a three-digit number where all digits are the same, such as 111, 222, ..., 999.
Value of \( a \)
- The three-digit number \( aaa \) can be expressed as:
\[
aaa = 111a
\]
- Thus, we can rewrite the equation as:
\[
x^2y = 111a
\]
Finding Possible Values
- Now, we need to find values of \( x \), \( y \), and \( a \) such that \( x^2y \) equals \( 111a \).
- Since \( 111 = 3 \times 37 \), \( a \) must be such that \( 1 \leq a \leq 9 \).
Testing Values of \( a \)
- Testing \( a = 1 \):
- \( 111a = 111 \), so \( x^2y = 111 \).
- Testing \( a = 2 \):
- \( 111a = 222 \), so \( x^2y = 222 \).
- Continue this for \( a = 3, 4, \ldots, 9 \).
Finding \( x, y \)
- For \( a = 3 \):
- \( x^2y = 333 \). Possible values could be \( (x, y) = (3, 37) \).
- For \( a = 4 \):
- \( x^2y = 444 \). Possible values could be \( (x, y) = (6, 12) \).
Calculating \( x + y + a \)
- For \( a = 3 \), \( x = 3 \) and \( y = 37\):
\[
x + y + a = 3 + 37 + 3 = 43
\]
- For \( a = 4 \), \( x = 6 \) and \( y = 12\):
\[
x + y + a = 6 + 12 + 4 = 22
\]
Conclusion
- The values of \( x+y+a \) can vary based on the valid pairs derived from \( a \). Thus, the possibilities include \( 22 \) and \( 43 \) based on tested values of \( a \).