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Please help with this number theory problem: Find the largest integer n such that 2^n divides 17^9 - 9^9.

 May 7, 2020
 #1
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I think the largest n = 3.

 

[17^9 - 9^9] / 2^3 =14,775,057,001

 May 7, 2020
 #2
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Please help with this number theory problem: Find the largest integer n such that \(2^n \)divides \(17^9 - 9^9\).

 

My attempt:

 

Formula:

\(\begin{array}{|rcll|} \hline a^9-b^9 &=& (a^3-b^3)(a^6+a^3b^3+b^6) \quad | \quad (a^3-b^3) = (a-b)(a^2+ab+b^2) \\ \mathbf{a^9-b^9} &=& \mathbf{(a-b)(a^2+ab+b^2)(a^6+a^3b^3+b^6)} \\ \hline \end{array}\)

 

\(\begin{array}{|rcll|} \hline \mathbf{a^9-b^9} &=& \mathbf{(a-b)(a^2+ab+b^2)(a^6+a^3b^3+b^6)} \\\\ 17^9-9^9 &=& (17-9)(17^2+17*9+9^2)(17^6+17^39^3+9^6) \\ 17^9-9^9 &=& 8*523*28250587 \\ && \boxed{523 ~\text{ is odd and }~ 28250587~\text{ is odd}\\ \text{and } 8=2^3 } \\ 17^9-9^9 &=& 2^3*523*28250587 \\ \hline \end{array} \)

 

The largest integer n is 3

 

laugh

 May 8, 2020

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