What should be added to the units digit of the number 86236 so that the resulting number is divisible by 9?

Introduction / Context:
This question tests your understanding of the divisibility rule for 9 and your ability to work with digit sums. To check if a number is divisible by 9, you do not need to divide it directly. Instead, you can sum its digits and see whether the sum is divisible by 9. Here, you are asked how much must be added to the units digit so that the entire number becomes divisible by 9.

Given Data / Assumptions:

• The original number is 86236.
• We are allowed to add a non-negative digit to the existing units digit (6) to create a new number.
• The new number must be divisible by 9.
• We must find the smallest such addition from the given options.

Concept / Approach:
The rule for divisibility by 9 states: A number is divisible by 9 if and only if the sum of its digits is divisible by 9. When we change the units digit by adding some value x, we effectively increase the sum of the digits by x. Therefore, the new sum should be a multiple of 9. We calculate the current sum of the digits, determine what x must be so that this sum plus x is divisible by 9, and then choose the appropriate option.

Step-by-Step Solution:
Step 1: Compute the sum of the digits of 86236. Step 2: 8 + 6 + 2 + 3 + 6 = 25. Step 3: Let x be the number to be added to the units digit. Then the new digit sum will be 25 + x. Step 4: For the new number to be divisible by 9, 25 + x must be a multiple of 9. Step 5: Note that 25 gives remainder 25 − 18 = 7 when divided by 9. So we need x such that 7 + x is a multiple of 9. Step 6: The smallest non-negative x between 0 and 9 that satisfies this is x = 2, because 7 + 2 = 9, and 9 is divisible by 9.

Verification / Alternative check:
If we add 2 to the units digit 6, the new number becomes 86238. The digit sum is 8 + 6 + 2 + 3 + 8 = 27. Since 27 is divisible by 9, 86238 is divisible by 9. This confirms that adding 2 to the units digit achieves the required property. Checking other small additions like 0 or 1 shows that the resulting digit sums (25 and 26) are not multiples of 9.

Why Other Options Are Wrong:
Option 1: 25 + 1 = 26, which is not divisible by 9.
Option 3: 25 + 3 = 28, which is not a multiple of 9.
Option 0: 25 + 0 = 25, also not divisible by 9.
Option 4: 25 + 4 = 29, again not divisible by 9.

Common Pitfalls:
Some students incorrectly try to divide the entire number by 9 each time they change the units digit, which is more time consuming and error prone. Others might miscalculate the digit sum. Using remainders helps: find the remainder of the digit sum when divided by 9, then choose x so that the new remainder becomes 0. This is a fast and reliable strategy for such problems.

Final Answer:
You must add 2 to the units digit so that the number becomes divisible by 9.