Understanding How Many 3-Digit Numbers Have a Ones Digit of 2, 4, or 6

In this article, we will explore how many three-digit numbers have a ones digit of 2, 4, or 6. This exercise is not only interesting but also helpful in understanding the multiplication principle and permutations. Let's delve into the detailed calculation and code implementation to find the solution.

Calculation Using the Multiplication Principle

To break down the problem into smaller parts, we will start by considering the range of three-digit numbers. Three-digit numbers range from 100 to 999. Now, let's consider the choices for each digit in these numbers:

Choices for the Ones Digit: The possible choices for the ones digit are 2, 4, or 6. This gives us 3 options. Choices for the Hundreds Digit: The hundreds digit can be any digit from 1 to 9 since it cannot be 0. This gives us 9 options. Choices for the Tens Digit: The tens digit can be any digit from 0 to 9. This gives us 10 options.

Now, we can calculate the total number of three-digit numbers with the specified condition using the multiplication principle:

[ text{Total} text{Choices for hundreds digit} times text{Choices for tens digit} times text{Choices for ones digit} ]

Substituting the numbers:

[ text{Total} 9 times 10 times 3 270 ]

Therefore, there are 270 three-digit numbers that have a ones digit of 2, 4, or 6.

Code Implementation to Verify the Solution

Let's verify this solution using Python. The code will generate all three-digit numbers with a ones digit of 2, 4, or 6 and then count them.

print(len([i for i in range(100, 1000) if str(i)[-1] in '246']))

Running the code will produce the following output:

270

This confirms that our initial calculation is correct.

Considering Negative Numbers

Now, let's extend our understanding to include negative numbers. For each positive number with a ones digit of 2, 4, or 6, there is a corresponding negative counterpart. For instance, if 102 is a valid number, -102 is also a valid number.

Since there are 270 positive three-digit numbers with a ones digit of 2, 4, or 6, there are also 270 negative numbers. Therefore, the total number of three-digit numbers (both positive and negative) with a ones digit of 2, 4, or 6 is:

[ text{Total} 270 270 540 ]

So, there are a total of 540 possible values from -102 to -996 and 102 to 996.

Conclusion

In conclusion, understanding how many three-digit numbers have a specific ones digit can be solved using the multiplication principle and verified through coding. The total count of such numbers is 270 for positive values and 540 when including negative values.

Keywords: 3-digit numbers, ones digit, counting numbers