Exploring 4-Digit Even Numbers Divisible by 5

Understanding how many 4-digit even numbers can be divisible by 5 is a fundamental concept in number theory and can provide insights into the properties of numbers. This article will walk you through the process and provide a detailed explanation of the mathematical principles involved.

Properties and Criteria

To find the number of 4-digit even numbers that are divisible by 5, we need to first establish the properties of such numbers. A 4-digit number is any number that lies between 1000 and 9999.

An even number has a last digit that is one of the following: 0, 2, 4, 6, or 8. On the other hand, a number is divisible by 5 if it ends in 0 or 5. Given this, the only possible last digit for a 4-digit even number divisible by 5 is 0, since 5 is an odd number and not an even.

Representing the Numbers

A 4-digit number can be represented as abc0, where:

a represents the thousands place and can be any digit from 1 to 9. b represents the hundreds place and can be any digit from 0 to 9. c represents the tens place and can also be any digit from 0 to 9.

Counting the Possibilities

Let us calculate the total number of such 4-digit numbers:

a has 9 possible values (1-9). b has 10 possible values (0-9). c has 10 possible values (0-9).

Hence, the total number of 4-digit even numbers that are divisible by 5 is:

$$text{Total numbers} 9 times 10 times 10 900$$

Brute Force Verification

To verify the result, we can use a brute force approach using the J programming language:

/05/ev 1000 to 9999900

The result confirms that the total number of 4-digit even numbers divisible by 5 is indeed 900.

Arithmetic Series Solution

An alternative method to verify this solution is through the arithmetic series formula. The first four-digit even number divisible by 5 is 1000, and the last one is 9990. The common difference between these terms is 10. The number of terms in an arithmetic sequence can be calculated as:

$$n frac{L - a_1}{d} 1$$

Where:

L is the last term (9990). a_1 is the first term (1000). d is the common difference (10).

Plugging in the values:

$$n frac{9990 - 1000}{10} 1 900$$

This confirms that there are 900 four-digit even numbers divisible by 5.

Summarizing the Results

To summarize, we have found that there are 900 four-digit even numbers that are divisible by 5. This includes only positive numbers. If we consider negative numbers as well, the total count would be 1800.

Understanding and applying these mathematical principles can help in various real-world applications and problem-solving scenarios, making it a valuable piece of knowledge for mathematicians and computer scientists.

For further reading or additional questions, feel free to explore more articles or consult a professional for deeper insights into number theory and its applications.