The Greatest 3-Digit Number Divisible by Both 3 and 9: A Comprehensive Guide
Understanding divisibility and the least common multiple (LCM) is crucial in many mathematical applications, particularly in number theory and problem solving. In this article, we will delve into the concept of the greatest 3-digit number that is divisible by both 3 and 9. We will explore the methods to find such numbers, emphasizing the significance of the least common multiple (LCM). By the end of this guide, you will have a clear understanding of how to identify the greatest 3-digit number that satisfies these criteria.
Introduction to Divisibility and LCM
Divisibility refers to the ability of one number to be evenly divided by another number without leaving a remainder. In mathematical terms, if a number (a) is divisible by (b), it means (a div b c) where (c) is an integer.
The least common multiple (LCM) of two or more numbers is the smallest positive integer that is divisible by each of them. This concept is particularly useful in solving problems related to time, ratios, and common multiples. For example, finding the LCM of 3 and 9 involves identifying the smallest number that is divisible by both numbers.
Understanding 3-Digit Numbers Divisible by 3 and 9
A 3-digit number divisible by both 3 and 9 must be a multiple of the least common multiple of 3 and 9. The smallest number that satisfies both conditions is the LCM of 3 and 9.
Find the LCM of 3 and 9
The prime factorization method can help us determine the LCM of 3 and 9. 3 31 9 32
The LCM is found by taking the highest power of each prime factor present in the factorizations:
LCM(3, 9) 32 9Since the LCM of 3 and 9 is 9, any number that is a multiple of 9 is automatically a multiple of both 3 and 9.
Identifying the Greatest 3-Digit Number Divisible by 9
The greatest 3-digit number is 999. To determine if 999 is divisible by 9, we perform the division:
999 รท 9 111
Since 999 divided by 9 results in an integer (111), we confirm that 999 is indeed the greatest 3-digit number divisible by 9.
Key Points to Remember
The greatest 3-digit number divisible by both 3 and 9 is 999. The LCM of 3 and 9 is 9. A number is divisible by 9 if the sum of its digits is divisible by 9.Conclusion
Understanding the concept of the least common multiple and its application in divisibility is a fundamental skill in mathematics. As demonstrated in this guide, the greatest 3-digit number divisible by both 3 and 9 is 999. This knowledge not only aids in solving specific mathematical problems but also enhances your overall problem-solving ability.
FAQs
What is the greatest 3-digit number divisible by 9?
The greatest 3-digit number divisible by 9 is 999. To verify, divide 999 by 9, which equals 111, an integer, confirming that 999 is a multiple of 9.
How do we find the LCM of 3 and 9?
The LCM of 3 and 9 is found by determining the prime factorizations and taking the highest powers of each prime. Since 3 31 and 9 32, the LCM is 32 9.
What is the significance of the least common multiple?
The LCM is significant because it helps identify the smallest number that is a multiple of two or more given numbers. This is particularly useful in various mathematical contexts, including time calculations, arithmetic operations, and simplifying fractions.
By following these guidelines, you will be well-prepared to tackle similar problems and deepen your understanding of mathematical concepts.