How to compute this greatest common divisor? I don't want the final answer, just a tip of how to begin, because I'm stuck in this problem.
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1Clearly the gcd will be even. – Michael Hardy Aug 03 '21 at 18:22
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2probably intended 5^222 – Will Jagy Aug 03 '21 at 18:22
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(Possibly a ) tip : I ...... guess you gotta divide and equate the GCD of the remainder and the divisor to that of the divisor and the dividend ? – Spectre Aug 03 '21 at 18:25
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As proved in the linked dupe, if $,b,c,$ are odd and coprime then $,\gcd(a^b+1,a^c+1) = a+1,,$ so $,\gcd(25^{101}+1,25^{37}+1) = 25+1\ $ – Bill Dubuque Aug 03 '21 at 18:40
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Note that $202-74 = 128$, so we can try to subtract off $5^{202}$ and reduce the problem like this:
\begin{align} &\gcd(5^{202} + 1, 5^{74} + 1) \\ =\,&\gcd(5^{202} + 1 - 5^{128}\cdot (5^{74} + 1), 5^{74} + 1) \\ =\,&\gcd(5^{202} + 1 - 5^{202} - 5^{128}, 5^{74} + 1) \\ =\,&\gcd(-5^{128} + 1, 5^{74} + 1) \end{align}
now you can continue reducing