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Chapter 2: Problem 1

If \(a \mid b\), show that \((-a)|b, a|(-b)\), and \((-a) \mid(-b)\).

Short Answer

Expert verified
All conditions hold as each can be expressed with respective integer multiples.

Step by step solution

01

Understand the Given Condition

We are given that \(a | b\). This means there exists an integer \(k\) such that \(b = ak\). We need to use this condition to prove three additional divisibility statements.
02

Prove that \((-a)|b\)

Since \(b = ak\), we can express \(b\) as \((-a)(-k)\). Since \(k\) is an integer, \(-k\) is also an integer. Therefore, \((-a)|b\) because \(b\) can be expressed as a product of \(-a\) and an integer.
03

Prove that \(a|(-b)\)

Given \(b = ak\), we know \(-b = -ak\). This expression shows that \(-b\) is a product of \(a\) and \(-k\) (an integer), so \(a|(-b)\).
04

Prove that \((-a)|(-b)\)

With \(-b = -ak\), we see \(-b = (-a)k\). Since \(k\) is an integer, \(-b\) can be expressed as a product of \(-a\) and \(k\). Thus, \((-a)|(-b)\).

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Key Concepts

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

Understanding Integers
Integers are foundational in mathematics. They include all the whole numbers, both positive and negative, as well as zero.
Examples of integers are -3, 0, 7, and -25. Unlike rational or irrational numbers, integers do not include fractions or decimals.
  • Positive integers: 1, 2, 3, ...
  • Negative integers: -1, -2, -3, ...
  • Zero: 0
Integers are crucial when discussing divisibility because they form the base of whole numbers.
If we say one integer divides another, it means the division results in an exact whole number with no remainder.
This concept plays a big part in understanding how the rules of mathematics apply to both positive and negative numbers.
Working with Negative Numbers
Negative numbers are integers that are less than zero. They represent a value below zero and can often be visualized on a number line as lying to the left of zero.
Calculating with negative numbers can sometimes be tricky, especially when it involves operations like division and multiplication.
  • Negative of a negative number is positive: For example, \(-(-3) = 3\).
  • Product of two negative numbers is positive:For example, \((-2) \times (-3) = 6\).
  • Product of a positive and a negative number is negative:For example, \(3 \times (-4) = -12\).
In terms of divisibility, understanding the role of negative numbers is crucial.
This is because when we multiply or divide by a negative number, sign changes affect the outcome. In exercises involving divisibility, make sure to manage these signs carefully, ensuring the integer relationships still hold.
Demystifying Mathematical Proof
A mathematical proof is a logical argument that establishes the truth of a statement. It consists of a series of logical steps that use previously established statements or axioms.
Proofs are important because they confirm that a particular mathematical statement is universally true.
  • Begin with known facts or conditions:In the given exercise, start with the fact that "\(a \mid b\)" means there exists an integer \(k\) such that \(b = ak\).
  • Follow logical steps:Break down the problem as seen in the solution, proving each required divisibility statement one step at a time.
  • Conclude with a validated statement:Each step logically follows from the previous, leading to a series of true statements, as shown in proving \((-a)|b\), \(a|(-b)\), and \((-a) \mid(-b)\).
When you work on proofs, make sure every step is clear and justified.
This clarity builds a chain of truth from the hypothesis all the way to the conclusion.

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Most popular questions from this chapter

Prove that no integer in the following sequence is a perfect square: $$ 11,111,1111,11111, \ldots $$ [Hint: A typical term \(111 \cdots 111\) can be written as $$ 111 \cdots 111=111 \cdots 108+3=4 k+3 .] $$

(a) Prove that if \(d \mid n\), then \(2^{d}-1 \mid 2^{n}-1\). [Hint: Use the identity $$ \left.x^{k}-1=(x-1)\left(x^{t-1}+x^{t-2}+\cdots+x+1\right) .\right] $$ (b) Verify that \(2^{35}-1\) is divisible by 31 and 127 .

When Mr. Smith cashed a check at his bank, the teller mistook the number of cents for the number of dollars and vice versa. Unaware of this, Mr. Smith spent 68 cents and then noticed to his surprise that he had twice the amount of the original check. Determine the smallest value for which the check could have been written. [Hint: If \(x\) denotes the number of dollars and \(y\) the number of cents in the check, then \(100 y+x-68=2(100 x+y) .]\)

The Catalan numbers, defined by $$ C_{n}=\frac{1}{n+1}\left(\begin{array}{c} 2 n \\ n \end{array}\right)=\frac{(2 n) !}{n !(n+1) !} \quad n=0,1,2, \ldots $$ form the sequence \(1,1,2,5,14,42,132,429,1430,4862, \ldots\). They first appeared in 1838 when Eug猫ne Catalan ( \(1814-1894)\) showed that there are \(C_{n}\) ways of parenthesizing a nonassociative product of \(n+1\) factors. [For instance, when \(n=3\) there are five ways: \(((a b) c) d,(a(b c)) d, a((b c) d), a(b(c d)),(a b)(a c) .]\) For \(n \geq 1\), prove that \(C_{n}\) can be given inductively by $$ C_{n}=\frac{2(2 n-1)}{n+1} C_{n-1} $$

If \(a\) and \(b\) are integers, not both of which are zero, verify that $$ \operatorname{gcd}(a, b)=\operatorname{gcd}(-a, b)=\operatorname{gcd}(a,-b)=\operatorname{gcd}(-a,-b) $$
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