Which of the following numbers should be added to 8567 to make it exactly divisible by 4?

### Concept & Rule The divisibility rule for 4 states that a number is perfectly divisible by 4 if and only if the number formed by its last two digits is divisible by 4. Therefore, we can ignore the "85" in 8567 and solely focus on making the number "67" divisible by 4. ### Step-by-Step Solution * **Step 1:** Extract the last two digits of the given number. The last two digits of $8567$ are $67$. * **Step 2:** Find the remainder when these last two digits are divided by 4. $67 \div 4 = 16$ with a remainder of $3$ (since $16 \times 4 = 64$). * **Step 3:** Determine the absolute minimum number to add. To reach the next multiple of $4$, we need to add: $(\text{Divisor} - \text{Remainder}) = 4 - 3 = 1$. Adding $1$ to $8567$ yields $8568$, which is divisible by $4$. * **Step 4:** Reconcile with the given options. The number $1$ is not in the options: (a) 3, (b) 4, (c) 5, (d) 6. However, if adding $1$ works, adding any subsequent multiple of $4$ will also work. General form of numbers to add $= 1 + 4k$ (where $k$ is an integer $0, 1, 2, ...$). If $k=1$, the number to add is $1 + 4(1) = 5$. This matches option (c). * **Step 5:** Verify. $8567 + 5 = 8572$. The last two digits are $72$. Since $72 \div 4 = 18$, it is perfectly divisible by 4. ### Exam Strategy & Shortcut **Direct Option Testing:** Just add the options directly to the last two digits ($67$) and check for divisibility by 4. (a) $67 + 3 = 70$ (Not div by 4) (b) $67 + 4 = 71$ (Odd, never div by 4) (c) $67 + 5 = 72$ ($72 \div 4 = 18$, Yes!) (d) $67 + 6 = 73$ (Odd) This takes less than 10 seconds. ### Common Pitfall Students calculate that $1$ is the least number to add. When they don't see $1$ in the options, they hastily select "(e) None of these". The question asks "Which of the following numbers...", meaning we must evaluate the specific options provided, not just look for the theoretical minimum. ### Final Answer **Therefore, the correct answer is 5.**