Problem 12 - Basic College Mathematics

Question

For Exercises \(1-16,\) answer yes or no and give a reason based on the tests for divisibility. Determine whether 9840 is divisible by \(2 .\)

Step-by-Step Solution

Answer

Yes, 9840 is divisible by 2 because its last digit is 0.

Step 1: Identify the Last Digit

To determine if 9840 is divisible by 2, first check the last digit of the number. According to the divisibility rule for 2, a number is divisible by 2 if its last digit is 0, 2, 4, 6, or 8.

Step 2: Check the Last Digit

Observe that the last digit of 9840 is 0.

Step 3: Apply Divisibility Rule

Since 0 is one of the digits that indicates divisibility by 2, 9840 is indeed divisible by 2.

Step 4: Conclusion

Based on the divisibility test, we can conclude that 9840 is divisible by 2.

Key Concepts

Divisibility by 2

Understanding the divisibility rule for 2 is simple and very important in mathematics. This rule helps us quickly determine whether a number can be divided by 2 without a remainder. To check if a number is divisible by 2, we only need to look at its last digit.
If the last digit is 0, 2, 4, 6, or 8, then the number is divisible by 2.

Let's apply this rule to the example number 9840.
- The last digit of 9840 is 0.
- Since 0 is one of the digits that indicate divisibility by 2, 9840 is divisible by 2.

Any number ending in 0 is automatically divisible by 2. Once you get the hang of checking the last digit, you'll find it easy to determine if any number is divisible by 2.

Mathematics Exercises

Practicing exercises that involve divisibility rules is crucial for mastering number properties. These exercises help build a strong foundation in mathematics. Always start with simple problems before moving on to more complex ones.

For instance, our task was to determine if 9840 is divisible by 2. By following a structured approach, you can find the solution step by step:
- Step 1: Identify the last digit of the number.
- Step 2: Check if the last digit fits the divisibility rule for 2 (which means it should be 0, 2, 4, 6, or 8).
- Step 3: If it fits the rule, then the number is divisible by 2.

Practicing such exercises not only helps you understand the concept deeply but also builds confidence in tackling more complex mathematical problems.

Number Properties

Number properties are fundamental concepts in mathematics. They include various rules and characteristics that numbers possess. Understanding these properties helps solve problems more efficiently.

Divisibility is a key property of numbers. When we say a number is divisible by another, it means that after division, there is no remainder.

For example, when determining if 9840 is divisible by 2, knowing the property that the last digit determines divisibility by 2 is very useful. This single property can save a lot of time and effort.

Here are some quick tips to remember:
- Divisibility by 2: Last digit is 0, 2, 4, 6, or 8.
- Divisibility by 3: Sum of the digits is divisible by 3.
- Divisibility by 5: Last digit is 0 or 5.

Mastering these properties makes math less daunting and more enjoyable!