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Understanding modular arithmetic is essential for solving problems like the one given where operations are performed 'modulo' a certain number. In modular arithmetic, numbers wrap around after they reach a certain value, known as the modulus. For instance, the expression \(7 \cdot 5 \mod 70\) can be visualized similar to the hours on a clock. If our clock were a '70-hour clock', we'd wrap around every 70 hours.

Remember that \(a \mod n\) is the remainder when \(a\) is divided by \(n\). In the given problem, multiplication within the system needs to be understood in the context of this modulo operation. The product \(7 \cdot 5\) would normally be 35, but since it's already less than 70, it stays 35 in the \(D_{70}\) system. When we add 7 to that result, we get 42 because \(42 \mod 70\) is simply 42, as it's also less than 70. Thus, modular arithmetic helps manage values within a cyclical range, which allows for the construction of various algebraic systems, such as the one indicated by \(D_{70}\).

Discrete Mathematics provides the foundational language and set of tools used to describe and solve a myriad of problems in computer science, information theory, and mathematical logic, to name a few areas. Unlike continuous mathematics, which deals with functions that can have an infinite number of values within intervals, discrete mathematics deals with countable, separate values.

In our exercise, we're dealing with distinct integers within a modular system, a staple concept in discrete mathematics. Boolean algebra, a part of discrete mathematics, uses binary variables and logical operations and is commonly used in computer circuit design and digital logic. The mistake in the original problem statement arises from a misunderstanding of the modular properties that govern the algebraic system in question – which is fundamentally based on rules of discrete mathematics.

Algebraic structures involve sets equipped with one or more operations that satisfy certain axioms. Sets like \(D_{70}\) in our exercise, are considered algebraic structures within the context of Boolean algebra. These structures allow mathematicians and computer scientists to work with different number systems and define rules for operations like addition and multiplication.

The operation within \(D_{70}\) gives it unique properties – in this case, a structure with modular arithmetic. Understanding how to perform operations in algebraic structures, such as calculating the products and sums modulo 70, is crucial for deriving the correct outcomes. The issue with the problem given is that it attempts to equate an expression that does not hold in the particular algebraic structure of \(D_{70}\), highlighting the importance of strictly adhering to the operational rules defined by any algebraic structure.