Problem 27 - Basic College Mathematics

Question

To answer Exercises \(25-32,\) consider the following numbers. \(\begin{array}{rrrr}56 & 200 & 75 & 35 \\ 324 & 42 & 812 & 402 \\ 784 & 501 & 2345 & 111,111 \\ 55,555 & 3009 & 2001 & 1005\end{array}\) Which of the above are divisible by \(5 ?\)

Step-by-Step Solution

Answer

75, 35, 55,555, 2005

Step 1: Review the divisibility rule for 5

A number is divisible by 5 if it ends in either 0 or 5.

Step 2: Examine each number

Check the last digit of each number from the provided list.

Step 3: Identify numbers ending in 0 or 5

Select the numbers that have either 0 or 5 as their last digit.

Step 4: List the numbers divisible by 5

The numbers are: 75, 35, 55,555, and 2005.

Key Concepts

Divisibility by 5

In mathematics, the concept of divisibility allows us to determine whether one number is divisible by another without performing full division. The divisibility rule for 5 is straightforward. A number is divisible by 5 if its last digit is either 0 or 5. This makes it easy to check quickly whether a number meets this criterion without any complicated calculations.
Let's go through a brief example. Consider the number 75. You only need to check if the last digit is 0 or 5. Since 75 ends in 5, it's divisible by 5. If the number were 76, it would not be divisible by 5 since it ends in 6. This rule saves a lot of time, especially when dealing with large lists of numbers like in the given exercise.

Mathematics solutions

Mathematical solutions often include specific methods and rules to arrive at an answer efficiently. In the context of divisibility, knowing the right rule simplifies the process a lot.
Following step-by-step methods can make problem-solving more accessible:

• First, understand the rule. In our case, a number must end in 0 or 5 to be divisible by 5.


• Next, apply the rule to each number in the list. You check the last digit of each number: 56, 200, 75, 35, and so on.


• Then, filter out the numbers that end in either 0 or 5.


• Finally, list those numbers. For the given exercise, the numbers that end in 0 or 5 are 75, 35, 55,555, and 1005. So, these are the numbers divisible by 5.

Working through problems methodically and using established rules not only makes solving them easier but also minimizes the chance of errors.

Number properties

Understanding number properties is crucial for grasping more complex mathematical concepts. The properties of numbers, such as even or odd characteristics, prime factors, and divisibility, form the foundation of various mathematical problems.
Divisibility is one such property. It helps determine whether a number can be evenly divided by another without leaving any remainder. Knowing these properties can aid in understanding and solving problems more efficiently.
For example, the divisibility rule for 5 is a property of numbers that makes it clear and simple to identify whether a number like 324 or 812 can be divided by 5. Since neither ends in 0 or 5, they are not divisible by 5. This property allows quick sorting and identification in large datasets, streamlining the problem-solving process.