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Advanced Factorization

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

Factorization of Integers

         

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

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

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

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

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

If \(a\), \(c\) and \(e\) are prime numbers such that \(a<c<e\) and \[3^{6}+3^{7}+3^{8}+3^{9}=a^b c^d e^f,\] what is the value of \(bdf\)?

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