Counting Numbers Divisible by 5 Between 1 and 400

Mathematics is a fascinating field that allows us to explore patterns and solve complex problems. One such problem involves finding the numbers between 1 and 400 that are divisible by 5. This article will guide you through the process of solving this problem, explaining not only the answer but also the reasoning behind it.

Understanding the Problem

We are looking for all the numbers within the range of 1 to 400 (inclusive) that are divisible by 5. To do this, we will use an arithmetic sequence to find the solution.

Arithmetic Sequence Approach

An arithmetic sequence is a sequence of numbers in which the difference between consecutive terms is constant. The general formula for the nth term of an arithmetic sequence is:

(a_n a_1 (n-1)d)

Where:

(a_1) is the first term of the sequence (d) is the common difference between consecutive terms (a_n) is the nth term of the sequence (n) is the number of terms in the sequence

Applying the Formula to the Problem

In our case, the first term (a_1 5), the common difference (d 5), and the last term (a_n 400). We need to find the number of terms, (n).

Using the given formula (a_n a_{n-1} d), we can rewrite it to solve for (n):

(400 5 (n-1) times 5)

Simplifying this, we get:

(400 5n - 5 5)

(400 5n)

Solving for (n):

(n frac{400}{5})

(n 80)

Conclusion

Therefore, there are a total of 80 numbers between 1 and 400 (inclusive) that are divisible by 5.

Additional Insights

This problem falls under the broader category of divisibility and number theory, which is a fundamental aspect of mathematical education. Understanding arithmetic sequences can also help in solving other types of problems, such as finding the common factors, multiples, and other sequences.

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By exploring and solving problems like this, you can enhance your mathematical skills and develop a deeper understanding of how numbers work.