Problem 26 - 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 \(2 ?\)

Step-by-Step Solution

Answer

The numbers divisible by 2 are: 56, 200, 324, 42, 812, 402, and 784.

Step 1: Identify the Rule for Divisibility by 2

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

Step 2: List the Numbers and Check the Last Digit

List each number provided and check the last digit to determine if it is even.

Step 3: Check Each Number

56 (last digit 6 - even), 200 (last digit 0 - even), 75 (last digit 5 - odd), 35 (last digit 5 - odd), 324 (last digit 4 - even), 42 (last digit 2 - even), 812 (last digit 2 - even), 402 (last digit 2 - even), 784 (last digit 4 - even), 501 (last digit 1 - odd), 2345 (last digit 5 - odd), 111,111 (last digit 1 - odd), 55,555 (last digit 5 - odd), 3009 (last digit 9 - odd), 2001 (last digit 1 - odd), 1005 (last digit 5 - odd).

Step 4: List the Numbers That Are Divisible by 2

From the above list, identify the numbers with even last digits: 56, 200, 324, 42, 812, 402, 784.

Key Concepts

divisibility by 2

Divisibility rules are simple shortcuts to identify if a number can be divided by another without performing the actual division. The rule for divisibility by 2 is straightforward and very useful in arithmetic.

To determine if a number is divisible by 2, you only need to check the last digit. If the last digit of a number is one of the even digits (0, 2, 4, 6, and 8), then the entire number is divisible by 2. For example, let's consider the numbers provided in the exercise:

• 56 has a last digit of 6 (even), so it is divisible by 2.
• 200 has a last digit of 0 (even), so it is divisible by 2.
• 75 has a last digit of 5 (odd), so it is not divisible by 2.
• 324 ends in 4, making it divisible by 2, and so on.

This simple rule can quickly tell you if a number is evenly divisible by 2 without performing any division.

even numbers

Understanding even numbers is essential for mastering the divisibility rule by 2. Even numbers are integers that can be exactly divided by 2. This means there is no remainder when an even number is divided by 2.

Key properties of even numbers include:

• They end in 0, 2, 4, 6, or 8. These digits indicate that the number can be divided by 2 without a remainder.
• They appear every other integer in the number line (2, 4, 6, 8, 10, etc.).

To put it simply, if a number ends in an even digit, it’s an even number and hence divisible by 2. In the provided exercise, this means that numbers like 56, 200, 324, 42, and others are easily identified as even. Knowing whether a number is even or odd is fundamental in many areas of math and helps simplify many arithmetic problems.

basic arithmetic

Basic arithmetic serves as the foundation of all math. The four main operations in arithmetic are addition, subtraction, multiplication, and division. Being fluent in these operations is crucial for solving more complex problems.

Focusing on division, especially divisibility, plays a key role in arithmetic. Divisibility helps determine whether one number is a multiple of another without performing detailed calculations. The rules of divisibility, such as for 2, act as shortcuts.

For example:

• In the exercise, identifying the last digits of numbers is a simple step under basic arithmetic. If the last digit is 0, 2, 4, 6, or 8, the number is divisible by 2.
• These simple checks reduce the amount of math you have to do, speeding up processes like factoring and simplifying fractions.

Mastering basic arithmetic, including quick rules like divisibility by 2, eases the path to more advanced math topics and makes arithmetic straightforward and efficient.