Counting Natural Numbers Divisible by 8 Between 23 and 100: A Guide for SEO and Mathematicians

Understanding how many natural numbers within a specific range are divisible by a particular number can be a useful mathematical skill, as well as important for SEO optimization. This article will walk you through the process of determining how many natural numbers between 23 and 100 are exactly divisible by 8. This guide is designed for both mathematicians and SEO practitioners to provide a clear and concise approach to solving this problem.

Step-by-Step Guide

First, let's break down the problem into manageable steps:

Step 1: Identify the first multiple of 8 greater than 23

To find the smallest multiple of 8 greater than 23, we can use the ceiling function:

lceil frac{23}{8} rceil lceil 2.875 rceil 3

Therefore, the first multiple is:

8 times 3 24

Step 2: Identify the largest multiple of 8 less than or equal to 100

To find the largest multiple of 8 less than or equal to 100, we use the floor function:

lfloor frac{100}{8} rfloor lfloor 12.5 rfloor 12

Therefore, the largest multiple is:

8 times 12 96

Step 3: List the multiples of 8 between 24 and 96

The multiples of 8 from 24 to 96 are:

24 32 40 48 56 64 72 80 88 96

Step 4: Count the multiples

The multiples form an arithmetic sequence:

First term, a 24 Common difference, d 8 Last term, l 96

We can use the formula for the n-th term of an arithmetic sequence:

l a (n-1) d

Rearranging to find n:

n frac{l - a}{d} 1

Substituting the values:

n frac{96 - 24}{8} 1 frac{72}{8} 1 9 1 10

Thus, there are 10 natural numbers between 23 and 100 that are exactly divisible by 8.

Understanding the Method

The method used here leverages the concept of arithmetic progression (AP). The formula for the n-th term of an AP is given by:

l a (n-1)d

Where:

a is the first term d is the common difference l is the last term n is the number of terms

In our example:

a 24 d 8 l 96

Rearranging the formula to solve for n gives:

96 24 (n-1)8

Simplifying:

72 8(n-1)

Therefore:

9 n-1

Thus:

n 10

So, there are 10 natural numbers between 23 and 100 that are exactly divisible by 8.

Putting It into Practice

To solve similar questions quickly, you can use the following simple hack:

Choose the nearest number divisible by 8 in the given range. For 23 to 100, the nearest lower number is 24 (8 * 3), and the nearest higher number is 96 (8 * 12). Calculate the total number of terms with the formula: 12 - 3 1 10

This method is efficient and can be applied to any range of numbers and divisors.

Conclusion

Understanding how to count natural numbers divisible by a particular number is a valuable skill for both mathematicians and SEO professionals. By using algebraic formulas and arithmetic progression, you can solve such problems efficiently. This knowledge can be leveraged to improve SEO strategies and content optimization for websites, ensuring that relevant content is optimized for search engines.

We hope this guide helps you in your mathematical and SEO endeavors. If you found this article helpful, please consider upvoting it!