Question

Even integer n is divisible by 3. What can you say about n?

Short answer

If an even number \(n\) is divisible by 3, it must be a multiple of 6.

Step-by-step solution

Understanding the properties of even numbers

An even number is any integer that can be expressed in the form \(2k\), where \(k\) is an integer. So \(n=2k\).

Establishing relation with divisibility by 3

The problem statement asserts that \(n\) is divisible by 3, this implies \(n=3m\) where \(m\) is another integer.

Determining the relation between \(k\) and \(m\)

Since \(n=2k\) and also \(n=3m\), it must be that \(2k=3m\). From this equation, it can be inferred that \(k\) must be a multiple of 3.

Conclusion

Considering the information from steps 2 and 3, it can be deduced that if \'n\' is an even number and is divisible by 3, then \'n\' must be a multiple of 6.

Key concepts

Even Numbers

Even numbers are a fundamental part of mathematics. An even number is any integer that can be rewritten in the form \(2k\), where \(k\) is an integer. This characteristic implies that when you divide an even number by 2, the result is another integer, further emphasizing the regular occurrence on the number line. A few examples include:

• 2, 4, 6, 8, and so forth

Understanding even numbers can help in operations like addition, subtraction, and especially divisibility because of their predictability. For instance, adding two even numbers always results in an even number, while subtracting two even numbers also gives you an even number.

Integer Properties

Integers are whole numbers that can be positive, negative, or zero. They are a broader category that includes even numbers and much more. Each integer has specific properties, such as:

• An integer multiplied by another integer gives an integer.
• Every integer is either even or odd.
• The set of integers is ordered, meaning you can always compare two integers to see which is bigger or smaller.

Understanding the properties of integers is crucial as it allows us to make deductions in number theory problems. For example, if \(n\) is an even integer such that \(n = 2k\), then we have used the property that multiplication of integers results in an integer, ensuring \(2k\) is also an integer.

Number Theory

Number theory is a branch of mathematics focused on the properties and relationships of numbers, especially integers. It explores concepts such as:

• Divisibility and factors
• Prime numbers
• Modular arithmetic

Consider the problem where an even integer \(n\) is divisible by 3: \(n = 3m\). This statement ties deeply into number theory as it involves divisibility, one of its core concepts. It guides us to understand that if \(n\) can be written as both \(2k\) and \(3m\), it is related to multiples of both 2 and 3, hence 6.

Multiples

Multiples refer to the product of a number and an integer. For instance, multiples of 2 are 2, 4, 6, 8, etc. In our example, where an even integer \(n\) is expressed as \(n = 2k\) and also \(n = 3m\), we discover \(n\) fits as a multiple of both 2 and 3. By number theory, if \(n = 2k = 3m\), it implies that \(k\) is a multiple of 3, then \(n\) is a multiple of 6.
Multiple uses and divisibility help reveal hidden structures and patterns within integers, aiding in problem-solving and proofs. Understanding how multiple concepts interlink gives insights not only into problems at hand but also into building more complex mathematical theories.