Anomalous Cancellation/Variants/37 + 13 over 37 + 24

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Theorem

$\dfrac {37 + 13} {37 + 24} = \dfrac {37^3 + 13^3} {37^3 + 24^3}$


Proof

\(\displaystyle 37^3 + 13^3\) \(=\) \(\displaystyle \paren {37 + 13}^3 - \paren {3 \times 37^2 \times 13 + 3 \times 37 \times 13^2}\)
\(\displaystyle \) \(=\) \(\displaystyle \paren {37 + 13}^3 - 3 \times 13 \times 37 \paren {37 + 13}\)
\(\displaystyle \) \(=\) \(\displaystyle \paren {37 + 13} \paren {\paren {37 + 13}^2 - 3 \times 13 \times 37}\)
\(\displaystyle \) \(=\) \(\displaystyle \paren {37 + 13} \paren {2500 - 1443}\)
\(\displaystyle \) \(=\) \(\displaystyle \paren {37 + 13} \times 1057\)


\(\displaystyle 37^3 + 24^3\) \(=\) \(\displaystyle \paren {37 + 24}^3 - \paren {3 \times 37^2 \times 24 + 3 \times 37 \times 24^2}\)
\(\displaystyle \) \(=\) \(\displaystyle \paren {37 + 24}^3 - 3 \times 24 \times 37 \paren {37 + 24}\)
\(\displaystyle \) \(=\) \(\displaystyle \paren {37 + 24} \paren {\paren {37 + 24}^2 - 3 \times 24 \times 37}\)
\(\displaystyle \) \(=\) \(\displaystyle \paren {37 + 24} \paren {3721 - 2664}\)
\(\displaystyle \) \(=\) \(\displaystyle \paren {37 + 24} \times 1057\)

Hence:

$\dfrac {37^3 + 13^3} {37^3 + 24^3} = \dfrac {1057 \paren {37 + 13} } {1057 \paren {37 + 24} } = \dfrac {37 + 13} {37 + 24}$

$\blacksquare$


Sources