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Home » The given sum can be simplified using partial fraction decomposition: - Sourci

The given sum can be simplified using partial fraction decomposition: - Sourci

📅 May 1, 2026 👤 scraface

May 01, 2026

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\gcd(5^{24} - 1, 5^{36} - 1) = 5^{12} - 1.
We now compute $ 5^{12} - 1 $:
5^2 = 25, \quad 5^4 = (25)^2 = 625, \quad 5^8 = 625^2 = 390625,
5^{12} = 5^8 \cdot 5^4 = 390625 \cdot 625.
We compute $ 390625 \cdot 625 $:
Break $ 625 = 600 + 25 $, so
\cdot 600 = 390625 \cdot 6 \cdot 100 = (2,343,750) \cdot 100 = 234,375...
\cdot 25 = (390625 \cdot 100)/4 = 39,062,500 / 4 = 9,765,625.
Add: $ 234,375,000 + 9,765,625 = 244,140,625 $.
So $ 5^{12} = 244,140,625 $, and

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