Counting Three-Digit Numbers Divisible by 2 or 3: A Comprehensive Guide

Determining how many three-digit numbers are divisible by 2 or 3 involves a systematic approach, best executed with the principle of inclusion-exclusion. This guide will walk you through the process, ensuring a deep understanding of the mathematical principles involved.

Introduction to the Problem

At its core, the problem asks us to find the total count of numbers between 100 and 999 (inclusive) that are either multiples of 2 or multiples of 3. This requires a multi-step approach, leveraging basic number theory and the inclusion-exclusion principle.

Step 1: Identifying the Range of Three-Digit Numbers

The range of three-digit numbers is straightforward, with the smallest being 100 and the largest 999. This forms the foundation for our subsequent calculations.

Step 2: Counting Multiples of 2

First, we count the multiples of 2 within the three-digit range. These numbers form an arithmetic sequence where:

First term (a) 100 Last term (l) 998 Common difference (d) 2

The formula for the nth term of an arithmetic sequence is given by:

l a (n-1)cdot d

Step 3: Counting Multiples of 3

Next, we count the multiples of 3. These numbers also form an arithmetic sequence. Here’s the breakdown:

First term (a) 102 (the smallest three-digit multiple of 3) Last term (l) 999 Common difference (d) 3

Applying the same formula:

l a (n-1)cdot d

Step 4: Counting Multiples of 6 (Common Multiples of 2 and 3)

The multiples of 6 within the range will help us avoid double-counting. These numbers form an arithmetic sequence as well:

First term (a) 102 (the smallest three-digit multiple of 6) Last term (l) 996 Common difference (d) 6

And, applying the formula once more:

l a (n-1)cdot d

Step 5: Applying the Principle of Inclusion-Exclusion

The principle of inclusion-exclusion allows us to find the total count without overcounting. We use the counts from steps 2, 3, and 4 to perform the final calculation:

Total Multiples of 2 Multiples of 3 - Multiples of 6

Total 450 300 - 150 600

This gives us the total count of three-digit numbers divisible by 2 or 3. Now, let's delve deeper into the mathematical implications and applications of this principle.

Further Implications and Applications

Understanding the principle of inclusion-exclusion is not just about counting numbers. It has broader applications in various fields such as combinatorics, probability, and even in solving real-world problems. For instance, if you were asked to count the number of three-digit numbers divisible by 2, 3, or 5, the principle would again be fundamental. The general approach remains consistent, but the arithmetic might become more complex due to the increased number of overlapping sequences.

In conclusion, the problem of finding how many three-digit numbers are multiples of 2 or 3 is a practical application of the principle of inclusion-exclusion. By breaking it down step by step, we can systematically solve it while enhancing our understanding of fundamental mathematical principles.