Problem 33 - 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 \(2 ?\)

Step-by-Step Solution

Answer

313,332, 876, 64,000, 7624, 1110, 9990, 111,126, 5128

Step 1 - Understand even and odd numbers

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

Step 2 - Check the last digit of each number

Look at the last digit of each number provided and determine if it is even.- 305 (last digit: 5)- 313,332 (last digit: 2)- 876 (last digit: 6)- 64,000 (last digit: 0)- 1101 (last digit: 1)- 7624 (last digit: 4)- 1110 (last digit: 0)- 9990 (last digit: 0)- 13,205 (last digit: 5)- 111,126 (last digit: 6)- 5128 (last digit: 8)- 126,111 (last digit: 1)

Step 3 - List the numbers with even last digits

From the above analysis, the numbers with even last digits are:- 313,332- 876- 64,000- 7624- 1110- 9990- 111,126- 5128

Key Concepts

even and odd numbers

Even and odd numbers are fundamental concepts in math. Let's break them down simply. An **even number** can be divided by 2 without leaving a remainder. For example, numbers like 2, 4, 6, and 8 are even.
On the other hand, an **odd number** leaves a remainder of 1 when divided by 2. Examples are 1, 3, 5, and 7.
Knowing whether a number is even or odd is essential because it helps determine its divisibility by 2. When solving problems about divisibility, always start by identifying if a number is even or odd.

divisible by 2

A number is considered **divisible by 2** if, when divided by 2, the result is a whole number with no remainder. Divisibility rules help us make quick calculations without performing lengthy division. Here are some key points to remember:

• A number divisible by 2 must be an even number.
• Any even number follows the rule and is instantly known to be divisible by 2.

For example, 10 is divisible by 2 because 10 ÷ 2 = 5 (without any remainder). In contrast, 7 is not divisible by 2 because 7 ÷ 2 = 3 with a remainder of 1. By using this quick mental check, you can easily determine if any number you encounter is divisible by 2.

last digit rule

The **last digit rule** is a handy shortcut in math to figure out divisibility. To see if a number is divisible by 2, you only need to look at its last digit. Here’s how it works:

• If the last digit is 0, 2, 4, 6, or 8, the number is divisible by 2.
• If the last digit is 1, 3, 5, 7, or 9, the number is not divisible by 2.

For instance, consider the number 876. The last digit is 6. Since 6 is an even number, 876 is divisible by 2. This trick saves you time and effort, especially with larger numbers. Check the last digit quickly, and you’ll know if the whole number can be evenly divided by 2.