Modular arithmetic helps in competitive programming by keeping numbers within manageable bounds, especially when dealing with large numbers or cyclic properties in mathematical problems.

Modular arithmetic is a key concept in competitive programming, especially when working with problems involving large numbers or cyclic properties. In many contests, you're required to work with large numbers that can easily exceed the limits of standard data types, leading to overflow. Modular arithmetic helps prevent this by reducing numbers to their remainder when divided by a fixed modulus, usually a prime number like 10^9 + 7. This allows you to perform arithmetic operations without worrying about overflow, as the result is always within the range of the modulus. Modular arithmetic is particularly useful in problems involving combinatorics, such as calculating large factorials or binomial coefficients, where the intermediate results can grow very large. By taking results modulo a prime number at each step, you can keep the numbers manageable while still arriving at the correct answer. Another common use of modular arithmetic is in problems involving cyclic properties, such as finding the remainder when dividing a number by a given modulus. For example, in problems involving circular arrays or sequences, modular arithmetic can help you efficiently wrap around indices. Understanding how to work with modular arithmetic, including techniques like modular exponentiation and the modular inverse, is essential for solving a wide range of problems in competitive programming.