Problem 4 - Algebra and Trigonometry with Analytic Geometry

Question

(a) an even number (b) a number divisible by 5 (c) an even number or a number divisible by 5

Step-by-Step Solution

Answer

An even number: divisible by 2. A number divisible by 5: ends in 0 or 5. An even number or divisible by 5: satisfies either condition.

Step 1: Define Even Numbers

An even number is a number that is divisible by 2 without a remainder. This means if a number \( n \) is even, it can be expressed in the form \( n = 2k \), where \( k \) is an integer.

Step 2: Identify Numbers Divisible by 5

A number is divisible by 5 if it ends in 0 or 5. In mathematical terms, a number \( m \) is divisible by 5 if it can be expressed as \( m = 5j \), where \( j \) is an integer.

Step 3: Apply the OR Condition

An 'or' condition means that a number satisfies the condition if it meets at least one of the criteria specified. Therefore, a number is either even or divisible by 5 if it fulfills the condition of either Step 1 or Step 2.

Key Concepts

Even Numbers

Even numbers are integers that can be perfectly divided by 2, leaving no remainder. They take the form of \( n = 2k \), where \( k \) is an integer. This implies that every even number is, essentially, a multiple of 2. Here are some key characteristics to help identify even numbers:

• The last digit of an even number is always one of the following: 0, 2, 4, 6, or 8.
• When you divide an even number by 2, the result is another integer.
• Examples include numbers like 4, 12, 18, and 200.

If you're examining more complex problems in number theory, knowing these traits will quickly help determine a number's evenness.
Recognizing even numbers is important because it simplifies calculations, especially when dealing with divisibility or other algebraic operations.

Divisibility Rules

Divisibility rules are shortcuts that help determine whether a number can be divided by another without actually performing the division. These rules are crucial when simplifying fractions and working with factors. Here we'll focus on the rule for divisibility by 5:

• A number is divisible by 5 if its last digit is either 0 or 5.
• Using divisibility rules saves time and effort when checking large numbers.
• This can be particularly useful in problems involving factors and multiples.

For example, checking if 275 is divisible by 5 is as easy as observing that it ends in a 5.
Understanding these rules aids not only in swift calculations but also in facilitating better comprehension of larger mathematical expressions and problem sets.

Mathematical Expressions

Mathematical expressions are combinations of numbers, variables, and arithmetic operations like addition, subtraction, multiplication, and division. They are the language of mathematics, allowing us to represent complex relationships in a concise form. Here's a breakdown of some aspects:

• Variables are symbols that stand in for unknown values.
• An expression can be something simple like \( x+2 \) or more complex such as \( 3x^2 + 2xy - y^2 \).
• Expressions can also include constants—values that do not change such as numbers.

The use of expressions is fundamental in creating equations, which can be solved to find specific values or describe relations between variables.
When managing equations, understanding mathematical expressions allows for more effective solving and graphing of mathematical problems.