Problem 68 - Fundamentals of Mathematics

Question

(Section 2.5) Determine if 41,826 is divisible by 2 and 3 .

Step-by-Step Solution

Answer

41,826 is divisible by both 2 and 3.

Step 1: Determine Divisibility by 2

To check if a number is divisible by 2, look at its last digit. If the last digit is even, then the number is divisible by 2. For 41,826, the last digit is 6, which is even. Therefore, 41,826 is divisible by 2.

Step 2: Determine Divisibility by 3

To check if a number is divisible by 3, calculate the sum of its digits and see if that sum is divisible by 3. First, find the sum of the digits: \[4 + 1 + 8 + 2 + 6 = 21\]Next, check if 21 is divisible by 3. Since 21 divided by 3 is exactly 7 with no remainder, 21 is divisible by 3. Therefore, 41,826 is divisible by 3.

Step 3: Conclusion

Since 41,826 is divisible by both 2 and 3, it satisfies the conditions given in the problem. Therefore, 41,826 is divisible by both 2 and 3.

Key Concepts

Divisibility by 2

To determine if a number is divisible by 2, there is an incredibly simple rule: just take a look at the number's last digit. This rule is effective because a number ends in one of the digits 0, 2, 4, 6, or 8, it is even. Therefore, the entire number can be divided evenly by 2 without leaving any remainder.
For example, let's consider the number 41,826 from the exercise. The last digit is 6. Since 6 is an even number, this makes 41,826 divisible by 2. This rule makes checking for divisibility by 2 quick and effortless without needing to perform complex calculations.

• Last digit is 0, 2, 4, 6, or 8
• Number is even ✅
• Divisible by 2 ✅

Divisibility by 3

Divisibility by 3 involves calculating the sum of a number's digits. If this sum itself is divisible by 3, then the original number is also divisible by 3. This rule is handy when dealing with large numbers, as it reduces everything down to a simple addition problem.

Let's see how this works with our example, 41,826. Break down the number into individual digits and add them together:
\[4 + 1 + 8 + 2 + 6 = 21\]
After finding the sum of 21, the next step is to determine if 21 is divisible by 3. When you divide 21 by 3, you get 7, without any remainder. This confirms that 21 is divisible by 3, and therefore, 41,826 is divisible by 3 as well.

• Add individual digits together
• If result is divisible by 3, so is the number ✅

Sum of Digits

The sum of digits concept is crucial for determining divisibility by certain numbers like 3. When you add up the digits of a number, you create a smaller number that is easier to check for divisibility. This is particularly valuable in preventing calculation errors that can arise when directly dividing large numbers.

In the exercise, we saw how adding 4, 1, 8, 2, and 6 resulted in the sum 21. This transformation from 41,826 to 21 simplifies checking divisibility with smaller numbers like 3.

This method is not only used for 3 but is an intuitive concept in mathematics that can extend to other applications and help students understand larger concepts like digit roots and optimization methods. By grasping the sum of digits, you leverage a powerful tool for simplification and clarity when tackling various math problems.

• Convert large numbers to manageable sums
• Helps in divisibility checks