Problem 13 - A Gentle Introduction to the Art of Mathematics

Question

Show that if two positive integers \(a\) and \(b\) satisfy \(a \mid b\) and \(b \mid a\) then they are equal.

Step-by-Step Solution

Answer

If \(a \mid b\) and \(b \mid a\), then \(a\) and \(b\) must be equal.

Step 1 - Understanding the Problem

The goal is to prove that if two positive integers, say \(a\) and \(b\), have the property that \(a\) divides \(b\) and \(b\) divides \(a\), then \(a\) and \(b\) must be equal.

Step 2 - Define the Conditions

Given that \(a \mid b\) and \(b \mid a\), it means there exist integers \(k\) and \(l\) such that \(a = bk\) and \(b = al\).

Step 3 - Substitute

Substitute \(b\) from the second equation into the first equation to get \(a = (al)k\). This simplifies to \(a = a(lk)\).

Step 4 - Simplify

Since \(a = a(lk)\), we can divide both sides by \(a\) (noting \(a\) is not zero) to get \(1 = lk\).

Step 5 - Analyze the Result

The only integers \(l\) and \(k\) that satisfy \(1 = lk\) for positive integers are \(l = 1\) and \(k = 1\). Therefore, \(a = b\).

Step 6 - Conclusion

Since we have shown that \(a = b\) when \(a \mid b\) and \(b \mid a\), we conclude that the integers \(a\) and \(b\) are equal.

Key Concepts

Positive Integers

Positive integers are numbers greater than zero that do not have fractions or decimal points. These numbers are often referred to as the counting numbers and include all natural numbers greater than zero, such as 1, 2, 3, and so on.
Understanding positive integers is crucial for many mathematical problems, including divisibility and proofs. They are foundational in number theory.
In the given exercise, both numbers under consideration, namely 'a' and 'b', are specified as positive integers. This ensures that these numbers are greater than zero and simplifies our calculations.

Divisibility

Divisibility is a concept in mathematics where one integer is divisible by another if the division yields a whole number. For example, 12 is divisible by 3 because 12 divided by 3 equals 4, which is an integer.
In the exercise, we are given that 'a' divides 'b' and 'b' divides 'a'. This can be noted as:

• a divides b: There exists an integer k such that b = a·k.
• b divides a: There exists an integer l such that a = b·l.

This property of mutual divisibility between 'a' and 'b' plays an essential role in proving that the two numbers are equal.

Mathematical Proof

Mathematical proof is a logical argument that establishes the truth of a mathematical statement. These proofs are critical in mathematics to confirm that certain properties or relationships consistently hold true.
The exercise asks us to prove that if two positive integers 'a' and 'b' satisfy the conditions that 'a' divides 'b' and 'b' divides 'a', then 'a' and 'b' must be equal. The proof involves the following steps:

1. Understanding the problem statement.
2. Defining the conditions as a = b·k and b = a·l for integers k and l.
3. Substituting one equation into the other.
4. Simplifying and analyzing the resulting equation.
5. Concluding the proof based on logical deduction.

This step-by-step approach ensures that the conclusion is based on rigorous logical reasoning.

Integer Properties

Integers possess various properties that are useful in mathematical proofs and problem-solving. Some key properties include:

• Closure: Sum or product of two integers is always an integer.
• Associativity: Grouping of integers in addition or multiplication does not affect the result.
• Identity: Adding zero or multiplying by one leaves the integer unchanged.
• Inverses: Each integer has an additive inverse that sums to zero.
• Distributivity: Multiplication distributes over addition.

Particularly important to this exercise is the concept of mutual divisibility.
We showed that if 'a' and 'b' are positive integers where 'a' divides 'b' and 'b' divides 'a', there are factors k and l such that '1 = k·l'. Since 'a' and 'b' are positive, k and l must also be positive.
The only way this equation holds for positive factors is if both k and l are 1, leading to the conclusion that 'a' equals 'b'. This showcases the importance of understanding and applying integer properties in proofs.