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641 | 2^32 + 1

Problem:
Show that \( 641 | 2^{32} + 1 \).

Solution:
(From Problem Solving Strategies, Arthur Engel)

\( 641 = 625 + 16 = 5^4 + 2^4 \).
So \( 641 | 2^{32} + 2^{28} \cdot 5^4 \).

Also, \( 641 = 640 + 1 = 2^7 \cdot 5 + 1\).
So \( 641 | (2^7 \cdot 5)^4 - 1 = 2^{28}\cdot 5^4 - 1 \).

Hence \( 641 | 2^{32} + 2^{28} \cdot 5^4 -(2^{28}\cdot 5^4 - 1) \). QED

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