Let's solve the problem of determining which digits can occupy the blank space in the number "12_" to make it divisible by 3.
### Steps for solving:
1. Understand the Divisibility Rule for 3:
A number is divisible by 3 if the sum of its digits is divisible by 3.
2. Examine the Given Number:
Here, we have the number "12_". The digits we know are '1' and '2'.
3. Calculate the Sum of Known Digits:
Sum of the digits '1' and '2':
[tex]\[
1 + 2 = 3
\][/tex]
4. Identify Possible Values for the Blank:
We need to find digits (0 through 9) that, when added to this sum (which is 3), result in a total that is divisible by 3.
5. Iterate through Possible Digits:
- Adding 0 to the sum:
[tex]\[
3 + 0 = 3 \quad \text{(divisible by 3)}
\][/tex]
- Adding 1 to the sum:
[tex]\[
3 + 1 = 4 \quad \text{(not divisible by 3)}
\][/tex]
- Adding 2 to the sum:
[tex]\[
3 + 2 = 5 \quad \text{(not divisible by 3)}
\][/tex]
- Adding 3 to the sum:
[tex]\[
3 + 3 = 6 \quad \text{(divisible by 3)}
\][/tex]
- Adding 4 to the sum:
[tex]\[
3 + 4 = 7 \quad \text{(not divisible by 3)}
\][/tex]
- Adding 5 to the sum:
[tex]\[
3 + 5 = 8 \quad \text{(not divisible by 3)}
\][/tex]
- Adding 6 to the sum:
[tex]\[
3 + 6 = 9 \quad \text{(divisible by 3)}
\][/tex]
- Adding 7 to the sum:
[tex]\[
3 + 7 = 10 \quad \text{(not divisible by 3)}
\][/tex]
- Adding 8 to the sum:
[tex]\[
3 + 8 = 11 \quad \text{(not divisible by 3)}
\][/tex]
- Adding 9 to the sum:
[tex]\[
3 + 9 = 12 \quad \text{(divisible by 3)}
\][/tex]
6. Find the smallest and greatest valid digits:
- The smallest digit that makes the number divisible by 3 is [tex]\(0\)[/tex].
- The greatest digit that makes the number divisible by 3 is [tex]\(9\)[/tex].
### Conclusion:
The smallest digit that can be placed in the blank space to make the number "12_" divisible by 3 is [tex]\(0\)[/tex], and the greatest possible digit is [tex]\(9\)[/tex].
