Base Equation Conversion

The problem asks us to find the number of possible integer solutions for the variables x and y in the given equation:

$$ (43)_x = (y3)_8 $$

This equation involves numbers represented in different number bases. To solve it, we need to convert both sides of the equation into the standard base 10 (decimal system).

Converting Numbers to Base 10

Recall that a number represented as $d_n d_{n-1} \dots d_1 d_0$ in base $b$ is equivalent to $d_n \times b^n + d_{n-1} \times b^{n-1} + \dots + d_1 \times b^1 + d_0 \times b^0$ in base 10.

Applying this conversion rule:

• Left side: $(43)_x$

  $$ (43)_x = 4 \times x^1 + 3 \times x^0 = 4x + 3 $$

• Right side: $(y3)_8$

  $$ (y3)_8 = y \times 8^1 + 3 \times 8^0 = 8y + 3 $$

Equation Relationship Derivation

By setting the base 10 values equal, we get:

$$ 4x + 3 = 8y + 3 $$

Subtracting 3 from both sides simplifies the equation to:

$$ 4x = 8y $$

Dividing both sides by 4 yields the direct relationship between $x$ and $y$:

$$ x = 2y $$

Base and Digit Constraints

In any number system, the base must be strictly greater than any digit used in that number. Also, digits must be non-negative integers.

Let's apply these rules:

• For the number $(43)_x$:
  • The digits are 4 and 3.
  • The base is $x$.
  • Since the base must be greater than the largest digit, we have the constraint: $x > 4$.
• For the number $(y3)_8$:
  • The digits are $y$ and 3.
  • The base is 8.
  • The base 8 is inherently greater than the digit 3.
  • The digit $y$ must be less than the base 8, so $y < 8$.
  • Digits must also be non-negative, so $y \ge 0$.
  • Combining these, the constraints for $y$ are $0 \le y < 8$.

Finding Possible Solutions

We are looking for integer values of $x$ and $y$ that satisfy both the equation $x = 2y$ and the constraints $x > 4$ and $0 \le y < 8$. We can systematically check integer values of $y$ within its allowed range (0 to 7).

Solution Count

The pairs $(x, y)$ that satisfy all the conditions are:

• $(6, 3)$
• $(8, 4)$
• $(10, 5)$
• $(12, 6)$
• $(14, 7)$

Counting these pairs, we find there are 5 possible solutions to the equation.