Question

Prove that if \(h / m\) and \(m / n\), and \(h / n\).

Short answer

The law of transitivity for division states that if a number h is divisible by m, and m in turn is divisible by n, then h is always divisible by n. Our proof confirmed this law; given that h divided by m and m divided by n lead us to the conclusion that h must divide by n.

Step-by-step solution

Explain divisibility

Divisibility is a property of integers. If an integer m divides another integer h, this means that there exists an integer k such that \(h = m * k\).

Express the given conditions

The problem states that \(h / m\) and \(m / n\). This means that there exist integers a and b such that \(h = m * a\) and \(m = n * b\). We need to prove that \(h / n\), or in other words, that there exists an integer c such that \(h = n * c\).

Substitute the second condition into the first

Substitute \(m = n * b\) into the equation \(h = m * a\) to get \(h = a * (n * b) = n * (a * b)\). Therefore, there exists an integer c (in this case \(c = a * b\)) such that \(h = n * c\). This proves the law of transitivity for division = 'If a number h is divisible by m, and m in turn is divisible by n, then h is always divisible by n.'

Key concepts

Mathematical Proofs

In mathematics, proving statements or theorems is a fundamental aspect that establishes the validity of assertions beyond any doubt. A mathematical proof consists of a logical series of statements, starting from known facts and axioms, leading to the conclusion we're trying to demonstrate. It's essentially a way of convincing why something is true.

When approaching divisibility, for instance, we use proofs to affirm the relationship between numbers with respect to division. Referring to our original exercise, we needed to confirm that if one number divides two others, there's a chain effect. Mathematical proofs in this context rely on clearly defined properties of integers and the structure of rational numbers. By following a rigorous step by step approach, as seen in the solution provided, we arrive at a conclusion that is universally accepted within the realm of mathematics.

Integer Properties

Integers are the set of whole numbers including zero, positive and negative numbers without any decimal or fractional parts. One of the fundamental properties of integers used in divisibility is closure under multiplication. This means that the product of any two integers will always be an integer.

As we observed in the exercise, this property allows us to express the relationship between divisible numbers using integers. If an integer 'm' divides another integer 'h', we can find an integer 'k' such that the equation \(h = m \times k\) holds true. These integer properties serve as the backbone for constructing proofs in divisibility problems because they give us reliable and predictable results. Understanding these intrinsic properties is key to grasp the meaning behind equations and to navigate through complex mathematical proofs.

Transitive Property

The transitive property is a logical rule that relates three or more elements in a specific way: if the first element relates to the second, and the second element relates to the third, then the first element must also relate to the third. In mathematics, this property is often applied to relations such as equality, order, and in our case, divisibility.

In the original exercise, the transitive property is illustrated through division. If 'h' is divisible by 'm', and 'm' is divisible by 'n', then based on the transitive property, 'h' must be divisible by 'n'. This logical leap isn't arbitrary; it relies on the unwavering structure of integer arithmetic. By substituting one divisible relationship into another as seen in the solution steps, we demonstrate the chain reaction set off by this fundamental property, solidifying its place as a critical concept in understanding mathematical relationships.