Find each of these values. a) (99^2 32)^3 15 b) (3^4 17)^2 11 c) (19^3 23)^2 31 d) (89^3 79)^4 2

step 1 Okay, so in this question we want to find a value of a 99 squared mod 32 cubed mod 15. So in thi

Find each of these values. a) (99^2 32)^3 15 b) (3^4 17)^2...

Find each of these values.
a) $\left(99^{2} \bmod 32\right)^{3} \bmod 15$
b) $\left(3^{4} \bmod 17\right)^{2} \bmod 11$
c) $\left(19^{3} \bmod 23\right)^{2} \bmod 31$
d) $\left(89^{3} \bmod 79\right)^{4} \bmod 26$

Key Concepts

Modular Arithmetic

Modular arithmetic involves performing calculations with integers where numbers wrap around upon reaching a fixed value called the modulus. This system focuses on the remainders after division and simplifies mathematical computations by reducing numbers modulo n, which is essential in number theory, cryptography, and computer science.

Modular Exponentiation

Modular exponentiation is the method of finding the remainder when an integer raised to an exponent is divided by a modulus. It is performed efficiently by reducing the intermediate results modulo n at each step, thereby preventing the explosion of large numbers. This technique is crucial for handling computations involving large exponents in a computationally manageable way.

Properties of Congruences

The properties of congruences allow arithmetic operations to be performed in the modular context. Fundamental rules like the preservation of addition, subtraction, and multiplication under congruence (if a ? b mod n, then a + c ? b + c mod n and a · c ? b · c mod n) enable the breakdown and simplification of complex modular expressions.

Exponentiation Rules

Exponentiation rules, such as the power of a power and the multiplication of exponents, are applicable within modular arithmetic to simplify expressions. By systematically applying these rules and integrating modular reduction at each step, one can manage nested exponents and streamline calculations that might otherwise be computationally intensive.

Fermat's Little Theorem

Fermat's Little Theorem states that if p is a prime number and a is not divisible by p, then a^(p-1) ? 1 (mod p). This theorem is extremely useful for reducing large exponents modulo prime numbers by allowing the exponent to be reduced modulo (p-1), thus simplifying many modular arithmetic problems when the conditions for the theorem are met.

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