Problem 36 - Basic College Mathematics

Question

To answer Exercises \(33-40\), consider the following numbers. \(\begin{array}{rrrr}305 & 313,332 & 876 & 64,000 \\ 1101 & 7624 & 1110 & 9990 \\\ 13,205 & 111,126 & 5128 & 126,111\end{array}\) Which of the above are divisible by \(5 ?\)

Step-by-Step Solution

Answer

305, 64,000, 1110, 9990, 13,205

Step 1 - Identify criteria for divisibility by 5

A number is divisible by 5 if its last digit is either 0 or 5.

Step 2 - List the given numbers

The given numbers are: 305, 313,332, 876, 64,000, 1101, 7624, 1110, 9990, 13,205, 111,126, 5128, 126,111.

Step 3 - Check each number's last digit

Examine the last digit of each number:1. 305 ends with 52. 313,332 ends with 23. 876 ends with 64. 64,000 ends with 05. 1101 ends with 16. 7624 ends with 47. 1110 ends with 08. 9990 ends with 09. 13,205 ends with 510. 111,126 ends with 611. 5128 ends with 812. 126,111 ends with 1

Step 4 - Determine which numbers are divisible by 5

Numbers from the list that end with 0 or 5 are divisible by 5. These numbers are:1. 3052. 64,0003. 11104. 99905. 13,205

Key Concepts

Divisibility by 5

Divisibility rules help us quickly determine if one number is divisible by another without performing long division.
For a number to be divisible by 5, its last digit must be either 0 or 5. This rule is simple but very useful.
For example, consider the number 305. Since it ends in 5, it meets the criterion and is divisible by 5.
Similarly, the number 64,000 ends with 0, making it also divisible by 5.
Exercises that ask you to determine if numbers are divisible by 5 will always follow this straightforward rule. Checking the last digit can save you a lot of time and effort.

Number Properties

Properties of numbers are fundamental in various mathematical operations and problem-solving scenarios.
Understanding number properties can help you predict behaviors and outcomes without extensive calculation.
For example, even and odd numbers have unique properties:

• Adding two even numbers or two odd numbers always gives an even number.
• Adding an even number to an odd number always results in an odd number.

Another essential property is the divisibility rule we used earlier. For a number to be divisible by 5, the last digit should be 0 or 5.
Recognizing these properties helps simplify problems and arrive at the solution more efficiently.

Step-by-Step Problem Solving

Breaking down problems into smaller, manageable steps is crucial in mathematics.
This method provides clarity and supports systematic thinking.
Let's revisit our problem about divisibility by 5 for a practical approach:

• Step 1: Identify the criteria for divisibility by 5 (last digit is either 0 or 5).
• Step 2: List the given numbers.
• Step 3: Examine the last digit of each number to check if they meet the criteria.
• Step 4: List the numbers that are divisible by 5.

By following these steps, you can logically and accurately determine the solution.
This method can be applied to various types of problems, making problem-solving easier and more structured.