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Chapter 13: Problem 25

Show that the sum of any three consecutive integers is always divisible by 3

Short Answer

Expert verified
The sum of any three consecutive integers \( n, n+1, \text{ and } n+2 \) is always divisible by 3 because it simplifies to \( 3(n+1) \).

Step by step solution

01

- Define the Consecutive Integers

Let's define the three consecutive integers as follows:Let the first integer be denoted as n. Then, the next two integers will be n+1 and n+2.
02

- Write the Sum of the Three Integers

The sum of the three consecutive integers is: \( n + (n+1) + (n+2) \)
03

- Simplify the Summation

Combine the terms to simplify the expression:\[ n + (n+1) + (n+2) = 3n + 3 \]
04

- Factor out the Common Multiple

In the expression \( 3n + 3 \), factor out the common multiple of 3:\[ 3(n + 1) \]
05

- Conclude the Divisibility

Since the sum of the three consecutive integers can be written as \( 3(n + 1) \), it is clear that the sum is always divisible by 3.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Consecutive Integers
Consecutive integers are numbers that follow one another in order. For example, 1, 2, and 3 are consecutive integers. Similarly, -2, -1, and 0 are also consecutive integers. In general, if we start with any integer n, the next consecutive integers will be n+1, n+2, n+3, and so on. Understanding this helps us build patterns and observe properties for numbers occurring in sequence. In our exercise, we defined three consecutive integers as n, n+1, and n+2.
Divisibility Rules
Divisibility rules are simple shortcuts to determine if a number is divisible by another without performing division. For instance:
- A number is divisible by 3 if the sum of its digits is divisible by 3
- A number is divisible by 2 if its last digit is even
In our exercise, we see that when we sum any three consecutive integers, i.e., n, n+1, and n+2, the result is always divisible by 3 because we can factor out a 3 from the sum. This kind of insight simplifies checking large computations.
Algebraic Expressions
Algebraic expressions allow us to represent numbers and their relationships using variables and operations. In our exercise, we formed the algebraic expression for the sum of three consecutive integers:
\[ n + (n + 1) + (n + 2) \]
We then simplified this to:
\[ 3n + 3 \]
Algebraic expressions are powerful tools in simplifying and solving equations, making it easier to recognize patterns and common factors in numerical problems.
Mathematical Proof
Mathematical proof is the process of demonstrating that a statement is true using logical reasoning and previously established statements such as theorems and axioms. In our exercise, the proof involves several steps:
1. Define the consecutive integers as n, n+1, and n+2.
2. Write their sum as an algebraic expression.3. Simplify to show that the sum can be factored.
4. Factor out 3 to demonstrate divisibility.
This structured approach ensures our conclusion is logically sound and irrefutable.

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Most popular questions from this chapter

Write a proof of the conjecture. Once you have completed the proofs, add the theorems to your list. The sum of the measures of the four angles of a quadrilateral is \(360^{\circ} .\) (Quadrilateral Sum Theorem)

Let \(2 n-1\) and \(2 m-1\) represent any two odd integers, and prove that the product of any two odd integers is always an odd integer.

Write a proof of the conjecture. Once you have completed the proofs, add the theorems to your list. In an isosceles triangle, the altitudes to the congruent sides are congruent. (Altitudes to the Congruent Sides Theorem)

For each statement, select always (A), sometimes (S), or never (N). a. An angle inscribed in a semicircle is a right angle. b. An angle inscribed in a major arc is obtuse. c. An arc measure equals the measure of its central angle. d. The measure of an angle formed by two intersecting chords equals the measure of its intercepted arc. e. The measure of the angle formed by two tangents to a circle equals the supplement of the central angle of the minor intercepted arc.

You have probably noticed that the sum of two odd integers is always an even integer. The rule \(2 n\) generates even integers and the rule \(2 n-1\) generates odd integers. Let \(2 n-1\) and \(2 m-1\) represent any two odd integers, and prove that the sum of two odd integers is always an even integer.
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