Introduction
Understanding the smallest three-digit number divisible by both 2 and 3 is a fundamental concept. This article explains the process step-by-step and provides a clear, comprehensive understanding. Knowing such a number is also important for understanding divisibility rules and least common multiples, which are crucial in various mathematical applications.
Divisibility Rule for 2 and 3
A number is divisible by 2 if it is even, and a number is divisible by 3 if the sum of its digits is divisible by 3. When a number is divisible by both 2 and 3, it means it is divisible by their least common multiple (LCM), which is 6.
Method 1: Starting from the Smallest Three-Digit Number (100)
Step 1: Identify the smallest three-digit number, which is 100.
Step 2: Check for divisibility by 2.
100 is even, so it is divisible by 2.
Step 3: Check for divisibility by 3.
The sum of the digits of 100 is (1 0 0 1). Since 1 is not divisible by 3, 100 is not divisible by 3.
Step 4: Advance to the next even number, which is 102.
102 is even, so it is divisible by 2.
102 is also divisible by 3, as the sum of its digits is (1 0 2 3), which is divisible by 3.
Thus, the smallest three-digit number divisible by both 2 and 3 is 102.
Method 2: Starting from Negative Numbers
The process is similar when starting from negative numbers, but for simplicity, let's stick to positive numbers. However, if we were to look at negative numbers, we would follow a similar approach.
Method 3: Using Knowledge of Divisibility by 6
Since 2 and 3's LCM is 6, a number divisible by both 2 and 3 must be divisible by 6. Starting from the smallest three-digit number (100), we divide it by 6:
100 ÷ 6 16 remainder 4.
Adding the remainder to 100, we get 100 6 - 4 102, which is divisible by both 2 and 3.
Conclusion
Therefore, the smallest three-digit number that is divisible by both 2 and 3 is 102. Using similar methods, you can verify that the largest three-digit number divisible by both 2 and 3 is 996 (since 999 ÷ 6 166 remainder 3, and 999 - 3 996).
Additional Insights and Keywords
Understanding the concept of divisibility by 2 and 3 helps in further exploring other mathematical properties and applications. This knowledge also contributes to a better grasp of number theory and the use of least common multiples in various mathematical contexts.