Divisibility Challenge: Find Missing Digits for 51— to Be Divisible by 10

To solve this problem, we'll focus on ensuring divisibility by 10 for the given number $51\text{ }_{—\text{ —}}$.

The divisibility rule for 10 states that a number must end with the digit 0 to be divisible by 10.

Thus, the missing digits should form a number that ends in 0. Given the partial number $51\text{ }_{1\text{ —}}$, we need to check potential combinations that satisfy this rule.

• Evaluation of the required final digit: Only the digit 0 satisfies the condition for divisibility by 10, meaning the complete number should be $51y0$ where $y$ is any digit.

Hence, the missing sequence to ensure divisibility by 10 is $1, 0$. Therefore, $51\text{ }_{1\text{ }0}$ is the solution.

Returning to the choices provided, the correct answer is choice 1: $1, 0$.

To conclude, the number is completed as $5110$, which is divisible by 10.