How many three digit natural numbers are exactly divisible by 4 without leaving any remainder?

Introduction / Context:
This aptitude problem checks your understanding of divisibility and counting of numbers in a range. Specifically, you must find how many three digit numbers are divisible by 4 exactly, with no remainder.

Given Data / Assumptions:
- Three digit numbers range from 100 to 999 inclusive.
- A number is divisible by 4 if it leaves remainder 0 when divided by 4.
- We need to count all such three digit numbers within the given range.

Concept / Approach:
To count multiples of a number in a range, we find the first multiple in the range, the last multiple in the range, and then use a simple counting formula. For any integer k, the number of multiples of k between m and n inclusive is floor(n / k) minus floor((m - 1) / k). Here k is 4, m is 100, and n is 999.

Step-by-Step Solution:
Step 1: Identify the first three digit number divisible by 4. 100 / 4 = 25 exactly, so 100 is a multiple of 4.Step 2: Identify the last three digit number divisible by 4. 999 / 4 gives quotient 249 with remainder, so the last multiple is 249 * 4 = 996.Step 3: The multiples form an arithmetic progression: 100, 104, 108, ..., 996 with common difference 4.Step 4: Number of terms in this progression = (last term - first term) / common difference + 1.Step 5: Compute: (996 - 100) / 4 + 1 = 896 / 4 + 1 = 224 + 1 = 225.Step 6: Therefore, there are 225 three digit numbers exactly divisible by 4.

Verification / Alternative check:
An alternative method uses floor functions directly. Count of multiples of 4 up to 999 is floor(999 / 4) = 249. Count of multiples of 4 up to 99 is floor(99 / 4) = 24. So the number of three digit multiples is 249 minus 24, which equals 225. This agrees with the arithmetic progression method, confirming the result.

Why Other Options Are Wrong:
Options 224, 223, and 222 are all smaller than the correct count and arise from typical minus one mistakes or errors in the formula. Option 226 is larger and might come from incorrectly including a non three digit multiple or misreading the range. Careful application of the formula shows that only 225 is correct.

Common Pitfalls:
Some students start from 104 instead of 100 or forget to include both endpoints. Others misapply the formula and use (996 - 100) / 4 without adding 1. Another error is to mix up the range and count numbers from 0 to 999 instead of 100 to 999.

Final Answer:
The number of three digit natural numbers that are exactly divisible by 4 is 225.