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Advanced factorization is a gateway to algebraic number theory, which mathematicians study in order to solve famous conjectures like Fermat's Last Theorem.

Evaluate \[\frac{(167 ^2-165 ^2)(163 ^2-161 ^2)(159 ^2-157 ^2)}{166 \times 162 \times 158}.\]

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Evaluate \[\sqrt{23 \times 24 \times 25 \times 26+1}.\]

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\(\quad\) What is the real number \(A\) that satisfies

\[\frac{77 ^4+77 ^2+1}{77 ^2+77+1}=78 ^2-A?\]

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What is the value of \[490000-\frac{490000 \times 490001+1}{490701}?\]

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