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toni07
- 25
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If a, b and m > 0 are integers such that a % b (mod m), then a^n % b^n (mod m) for all positive integers n. I don't know how to go about it, any help would be greatly appreciated.
Congruence modulo m is a mathematical concept that describes the relationship between two numbers when they have the same remainder when divided by m. In other words, two numbers are congruent modulo m if they have the same remainder when divided by m.
To prove the operations of congruence modulo m, we use the properties of modular arithmetic such as the distributive, associative, and commutative properties. We also use the fact that two numbers are congruent modulo m if and only if their difference is divisible by m.
Congruence modulo m and equality are two different mathematical concepts. Congruence modulo m compares the remainders of two numbers when divided by m, while equality compares the values of two numbers. Congruence modulo m is a more specific and restricted relationship between two numbers compared to equality.
No, congruence modulo m is only defined for integer numbers. This is because the concept of congruence involves dividing the numbers and looking at their remainders, which is not possible for non-integer numbers.
Congruence modulo m is used in cryptography to encrypt messages and ensure their security. It is used in algorithms such as the RSA algorithm, which relies on the difficulty of factoring large numbers. The concept of congruence modulo m plays a crucial role in the mathematical foundations of these algorithms.