Problem 110 - Fundamentals of Mathematics

Question

(Section 2.5) Is 634,281 divisible by 3 ?

Step-by-Step Solution

Answer

Yes, 634,281 is divisible by 3.

Step 1: Understand Divisibility Rule for 3

A number is divisible by 3 if the sum of its digits is divisible by 3. Therefore, we need to find the sum of all the digits of the number 634,281 to determine its divisibility by 3.

Step 2: Calculate the Sum of the Digits

Add up all the digits of the number 634,281. This means we need to calculate: 6 + 3 + 4 + 2 + 8 + 1.

Step 3: Perform the Addition

Calculate the result of the addition: 6 + 3 + 4 + 2 + 8 + 1 = 24.

Step 4: Check if the Sum is Divisible by 3

Now we need to check if 24 is divisible by 3. Since 24 divided by 3 equals 8 without any remainder, 24 is divisible by 3.

Step 5: Conclude Divisibility

Since the sum of the digits, 24, is divisible by 3, we can conclude that the original number, 634,281, is also divisible by 3.

Key Concepts

Divisibility by 3

To determine if a number is divisible by 3, we can use a simple yet effective rule based on its digits. Instead of dividing large numbers directly, we can sum up all the digits of the number and see if this total is divisible by 3.

This rule works because of how numbers are structured in the base-10 number system. When digits are summed and remain divisible by 3, the whole number follows this rule, saving time and effort. Here's a simplified process to apply this rule:

• Add all the digits of the number together.
• Check if the resulting sum can be divided evenly by 3, meaning no remainder.

This method not only simplifies calculations but also strengthens understanding of basic mathematical properties.

Sum of Digits

The sum of digits involves adding together all individual digits of a number. This is an essential step in checking divisibility by 3. Let's see how it works using the given number 634,281 as an example:

1. Break down the number to its separate digits: 6, 3, 4, 2, 8, and 1.
2. Add them up:

• 6 + 3 = 9
• 9 + 4 = 13
• 13 + 2 = 15
• 15 + 8 = 23
• 23 + 1 = 24

Now that we've arrived at the sum of 24, we proceed to check this total's divisibility by 3. Understanding this step makes divisibility checks much easier, particularly when dealing with large numbers.

Moreover, it helps practice basic addition and enhances number sense.

Mathematical Concepts

Delving into the divisibility rules offers a glimpse into some fundamental mathematical concepts. It's not just about following a procedure but also about understanding why it works. Learning and applying the rule for divisibility by 3 enhances logical thinking and computational skills.

Through practicing with this rule, students can:

• Gain insights into the properties of numbers.
• Learn to manipulate numbers to simplify complex problems.
• Improve mental math capabilities.

Moreover, concepts like breaking down numbers, identifying patterns, and using rules like divisibility are foundational skills in many areas of mathematics and everyday problem-solving. Recognizing how interconnected mathematical ideas are can deepen appreciation and encourage further exploration in the subject.