Problem 65 - Algebra 2

Question

If \(a\) and \(b\) are natural numbers, then which of the following must also be a natural number? $$ \begin{array}{lll}{\text { I. } a-b} & {\text { II. } a b} & {\text { III. } \frac{a}{b}} \\ {\text { A I only }} & {\text { C III only }} \\ {\text { B II only }} & {\text { D I and II only }}\end{array} $$

Step-by-Step Solution

Answer

II only (Option B is correct).

Step 1: Understand Natural Numbers

Natural numbers are positive integers starting from 1, 2, 3, and so on. They do not include zero or any negative numbers.

Step 2: Analyze Statement I

Statement I suggests that \(a-b\) must be a natural number. If \(a\) is less than \(b\), then \(a-b\) will be negative, and negative numbers are not considered natural numbers. Thus, \(a-b\) is not necessarily a natural number.

Step 3: Analyze Statement II

Statement II suggests that \(ab\) must be a natural number. Since both \(a\) and \(b\) are natural numbers, their product is also a natural number. Hence, \(ab\) is always a natural number.

Step 4: Analyze Statement III

Statement III suggests that \(\frac{a}{b}\) must be a natural number. For this to be true, \(a\) must be a multiple of \(b\). However, in general, division of natural numbers does not necessarily result in a natural number unless specified conditions (\(b\) divides \(a\) exactly) are met. Therefore, \(\frac{a}{b}\) is not always a natural number.

Step 5: Determine Correct Answer

From the analysis, only Statement II can be concluded to be true under all conditions given \(a\) and \(b\) are natural numbers. Hence, only \(ab\) is always a natural number.

Key Concepts

Positive Integers

Natural numbers are a set of positive integers starting from 1 and increasing indefinitely: 1, 2, 3, 4, and so forth. These numbers are essential in the field of mathematics. They are used to represent quantities that cannot be divided into smaller units. They do not include zero, negative numbers, or fractions.
Understanding positive integers is crucial because they are fundamental building blocks for more advanced mathematical operations.

• Natural Numbers: These are the simplest form of numbers used for counting.
• Keep it Positive: Natural numbers always remain positive, making them intuitive for counting and measuring.

Integer Multiplication

Integer multiplication involves multiplying two or more integers together to get a product. When you multiply two natural numbers (positive integers), the result is always another natural number. This property is because the product of any two positive numbers remains positive.
Multiplication is an essential arithmetic operation, and it builds on the concept of repeated addition.

• Closed Under Multiplication: The set of natural numbers is closed under multiplication. This means multiplying any two natural numbers will always give a natural number.
• Commutative and Associative: Multiplication is commutative (i.e., a * b = b * a) and associative ((a * b) * c = a * (b * c)), which helps in simplifying complex calculations.

Subtraction Rules

Subtraction of integers can sometimes lead to results that are not natural numbers, especially when dealing with natural numbers. Consider the example where you subtract a larger number from a smaller one; the result will be negative or zero, which are not natural numbers.
Natural numbers do not include negative values, hence caution is needed when subtracting.

• Order Matters: Unlike addition, subtraction is not commutative (i.e., a - b ≠ b - a).
• Condition for Natural Number: For a subtraction operation (a - b) to result in a natural number, it is necessary that a ≥ b.

Integer Division

Integer division involves dividing one integer by another. In the context of natural numbers, this division does not always result in another natural number. For \( \frac{a}{b} \) to be a natural number, \(a\) must be exactly divisible by \(b\) with no remainder.
However, this condition is not always met.

• Exact Division: Only when the divisor divides the dividend exactly (i.e., the remainder is zero), can the quotient be a natural number.
• Not Always Natural: If \(b\) does not divide \(a\) evenly, the quotient becomes a fraction or decimal, which is not a natural number.