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In mathematics, the concept of remainders is crucial when discussing division and modular arithmetic. When you divide one number by another, there are often leftovers. These leftovers are what we call remainders.
For example, when you divide 14 by 3, you get 4 with a remainder of 2. This is because 3 goes into 14 four times (12), leaving 2 as the remainder. This remainder is what the modulus operation looks for.
In the context of the exercise, the modulus operation helps us find the remainder when each number from 0 to 14 is divided by both 3 and 5. The remainder is crucial for forming the pairs \((a \bmod 3, a \bmod 5)\).
Understanding remainders helps us better grasp how numbers break down into smaller, manageable parts, which is a fundamental aspect of number theory.

Integer division is a type of division in which the result is adjusted to the nearest integer, bypassing any decimal values. Instead of a precise quotient, integer division gives you just the whole number part.
For instance, dividing 14 by 3 in integer division gives you 4 as the result, ignoring the decimal part (4.666...). The leftover part is handled by the modulus operation, which provides the remainder.
In the given exercise, integer division is inherently involved in understanding how often one number can fit into another before you're left with the remainder. When working with \((a \bmod 3)\) and \((a \bmod 5)\), you repeatedly perform a kind of integer division where you're concerned with the leftover, not the exact quotient.
This process is useful in various fields including computer science, where integer division and modulus operations are often used for algorithms and data structures.

Number theory is a branch of mathematics focused on the properties and relationships of numbers, especially integers. One of the main topics in number theory is modular arithmetic, which deals with remainders.
Modular arithmetic is like clock arithmetic. For example, if it's 11 o'clock now, 5 hours later it will be 4 o'clock. Here, a modulus operation is used to wrap around numbers within a set range (like hours on a clock).
In this exercise, you're using number theory principles to find pairs of remainders for numbers 0 through 14 when divided by 3 and 5. This allows you to see patterns and relationships between numbers, which is a key aspect of number theory.
Understanding these fundamental principles can help you solve more complex problems in algebra, cryptography, and computer science, where modular arithmetic and number theory are extensively used.