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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.

# 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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