Highest Value X Calculation

The problem requires finding the maximum possible digit value for X in the given equation $5\text{Y}6 + 6\text{X}7 + 3\text{Z}8 = 1511$. Here, X, Y, and Z represent single digits from 0 to 9.

Equation Structure Analysis

The equation represents the sum of three three-digit numbers. We can express these numbers in terms of their place values:

• $5\text{Y}6 = 5 \times 100 + \text{Y} \times 10 + 6$
• $6\text{X}7 = 6 \times 100 + \text{X} \times 10 + 7$
• $3\text{Z}8 = 3 \times 100 + \text{Z} \times 10 + 8$

Digits Relationship Derivation

Let's analyze the sum column by column to determine the relationship between the digits X, Y, and Z:

1. Units Column: The sum is $6 + 7 + 8 = 21$. The units digit of the result (1) is correct. A carry-over of 2 is generated for the tens column.
2. Tens Column: The sum in this column is $\text{Y} + \text{X} + \text{Z} + (\text{carry-over } 2)$. This sum must equal the tens digit of the result (1) plus 10 times the carry-over ($C_h$) to the hundreds column. Therefore, $\text{Y} + \text{X} + \text{Z} + 2 = 1 + 10 \times C_h$.
3. Hundreds Column: The sum in this column is $5 + 6 + 3 + C_h$. This sum must equal the hundreds digit of the result (5) plus 10 times the carry-over to the thousands column. Since the total sum is 1511, the carry-over to the thousands column is 1. So, $5 + 6 + 3 + C_h = 5 + 10 \times 1$. $14 + C_h = 15$. Solving for $C_h$, we get $C_h = 1$.

Substitute $C_h = 1$ back into the tens column equation:

$\text{Y} + \text{X} + \text{Z} + 2 = 1 + 10 \times 1$

$\text{X} + \text{Y} + \text{Z} + 2 = 11$

$\text{X} + \text{Y} + \text{Z} = 9$.

X Value Maximization

We found that $\text{X} + \text{Y} + \text{Z} = 9$. To maximize X, we need to minimize the values of Y and Z. Since Y and Z are digits, their minimum possible value is 0.

• Set $\text{Y} = 0$.
• Set $\text{Z} = 0$.

Substitute these minimum values into the sum equation:

$\text{X} + 0 + 0 = 9$

$\text{X} = 9$.

The highest possible value for X is 9. This is achieved when Y=0 and Z=0, resulting in the valid equation $506 + 697 + 308 = 1511$.