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Find ${5^{2009}}^{1492}\mod{503}.$
How do you calculate a beast like this?
How do you calculate a beast like this?
MHB Can you simplify a monstrous remainder problem using modular arithmetic?
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To calculate ${5^{2009}}^{1492}\mod{503}$, one can utilize modular arithmetic, specifically Euler's theorem and Fermat's little theorem. Since 503 is prime, the period of powers of 5 modulo 503 is 502, derived from 503 - 1. Simplifying the exponent involves finding ${2009 \cdot 1492} \mod{502}$ to reduce the complexity of the expression. Analyzing a smaller modulus, such as 7, can also help clarify the reasoning process. Understanding these concepts is crucial for tackling large exponentiation problems in modular arithmetic.
Obviously, there is something elementary I am missing here.
To form the transpose of a matrix, one exchanges rows and columns, so the transpose of a scalar, considered as (or isomorphic to) a one-entry matrix, should stay the same, including if the scalar is a complex number. On the other hand, in the isomorphism between the complex plane and the real plane, a complex number a+bi corresponds to a matrix
in the real plane; taking the transpose we get
which then corresponds to a-bi...
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