If the number 583_437 is completely divisible by 9, then what is the smallest whole digit that can be placed in the blank position (represented by _) to make the number divisible by 9?

Introduction / Context:
This question tests the divisibility rule for 9 and asks you to determine a missing digit in a large number so that the entire number is divisible by 9. Among all such digits that work, you must choose the smallest one.

Given Data / Assumptions:
The number is 583_437, where the blank _ is a single digit from 0 to 9.The complete number must be divisible by 9.We must find the smallest digit that can replace the blank.

Concept / Approach:
A number is divisible by 9 if and only if the sum of its digits is a multiple of 9. We first compute the sum of the known digits in 583_437, then add the unknown digit x. We then find values of x that make the total a multiple of 9, and select the smallest such value.

Step-by-Step Solution:
Step 1: List the digits of 583_437: 5, 8, 3, x, 4, 3, 7.Step 2: Compute the sum of the known digits: 5 + 8 + 3 + 4 + 3 + 7.Step 3: Add these: 5 + 8 = 13, 13 + 3 = 16, 16 + 4 = 20, 20 + 3 = 23, 23 + 7 = 30.Step 4: The total digit sum is 30 + x.Step 5: For divisibility by 9, 30 + x must be a multiple of 9.Step 6: Multiples of 9 close to 30 are 27 and 36. Since 30 + x must be at least 30, the first possible multiple is 36.Step 7: Set 30 + x = 36 to get x = 6.Step 8: Another possible multiple would be 45, giving x = 15, which is not a single digit, so it is invalid.Step 9: Therefore, the only valid choice for x is 6, which is also the smallest possible.

Verification / Alternative check:
Check the full number 5836437. The digit sum is 5 + 8 + 3 + 6 + 4 + 3 + 7 = 36. Since 36 is a multiple of 9, the number is divisible by 9. No smaller digit x from 0 to 5 gives a total digit sum that is a multiple of 9.

Why Other Options Are Wrong:
If x = 3, the sum would be 33, which is not a multiple of 9. If x = 4, the sum would be 34, and if x = 5, the sum would be 35, neither of which is divisible by 9. Only x = 6 gives sum 36, satisfying the divisibility rule.

Common Pitfalls:
Some learners mistakenly look for divisibility by 3 instead of 9, or forget that the blank must be a single digit. Others incorrectly add the digits or stop at the first number they find without checking whether it is a multiple of 9. Being systematic with the digit sum and checking nearest multiples of 9 avoids these errors.

Final Answer:
The smallest digit that can replace the blank is 6.