Compute Modulo by Modding First

Compute $(45 \times 38) \mod 7$ by modding first.

In this problem, we explore the concept of modular arithmetic, which is a core topic within number theory. When you encounter a problem like computing (45 times 38) modulo 7, an efficient strategy involves reducing each number modulo 7 before performing the multiplication. This step is known as 'modding first' and takes advantage of the properties and rules of modular arithmetic. Specifically, since modular arithmetic is concerned only with the remainder of a division, modding first can simplify calculations by working with smaller, more manageable numbers. This approach uses the distributive property of multiplication over addition in modular mathematics, allowing you to maintain equivalency throughout the problem while simplifying computations.

Modular arithmetic appears in various areas of mathematics and computer science, particularly in problems involving periodicity or cyclical behavior. It forms the foundation of various algorithms in cryptography, computer science, and algorithms. By practicing problems with modular arithmetic, you're not only learning how to handle remainders but also preparing for topics like cryptographic hash functions and error detection protocols.

This problem, though relatively straightforward in computational terms, introduces the consistent use of modular arithmetic rules, offering a stepping stone to more complex scenarios. The skills honed here will be crucial when you deal with algorithms that implement modular reductions as part of larger computational processes, like the Euclidean algorithm or finding multiplicative inverses.