Problem 9 - Basic College Mathematics

Question

For Exercises \(1-16,\) answer yes or no and give a reason based on the tests for divisibility. Determine whether 21,687 is divisible by 2 .

Step-by-Step Solution

Answer

No, 21,687 is not divisible by 2 because its last digit (7) is not even.

Step 1: Understand the Divisibility Rule for 2

A number is divisible by 2 if its last digit is even (i.e., 0, 2, 4, 6, or 8).

Step 2: Check the Last Digit

Look at the last digit of the number 21,687, which is 7.

Step 3: Apply the Rule

Since 7 is not an even number, 21,687 is not divisible by 2.

Step 4: Conclusion

Based on the rule and the observation, we can conclude that 21,687 is not divisible by 2.

Key Concepts

Elementary Mathematics

In elementary mathematics, understanding basic rules and principles is crucial for solving problems effectively. One important concept is the use of divisibility rules, which helps simplify and solve problems more quickly.
These rules are simple guidelines that tell us if one number can be divided by another without having to do the division. For example, knowing if a number is divisible by 2, 3, 5, or any other number helps in various arithmetic operations like simplifying fractions and finding factors.
Understanding these basic concepts lays a solid foundation for more advanced topics in mathematics, such as algebra and calculus.

Divisibility by 2

The rule for divisibility by 2 is straightforward: A number is divisible by 2 if its last digit is even.
Even digits include 0, 2, 4, 6, and 8.
For example, the numbers 24, 102, and 78 all end in even numbers, making them divisible by 2.
In the given exercise to determine if 21,687 is divisible by 2, we only need to look at the last digit, which is 7. Since 7 is not an even number, 21,687 is not divisible by 2. This rule saves us time and effort, preventing the need for long division to determine divisibility.

Number Theory

Number theory is a branch of mathematics that studies the properties and relationships of numbers, particularly integers. One area of focus in number theory is the exploration of divisibility and the characteristics of prime and composite numbers.
Divisibility rules are a practical application within number theory, helping us quickly identify factors and multiples without elaborate calculations.
These rules, including the one for divisibility by 2, are based on patterns and properties of numbers.
Using these rules can help solve more complex problems related to factors, multiples, and divisors.
By understanding and applying these fundamental principles, students build a stronger mathematical foundation and develop critical thinking skills for higher-level mathematics.