rational
  • rational
Find the \(\gcd\) of the set of numbers \[\{16^n+10n-1|n=1,2,3,\ldots\}\]
Discrete Math
  • Stacey Warren - Expert brainly.com
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SOLVED
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katieb
  • katieb
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rational
  • rational
I see that \(25\) is common by evaluating the given expression for first few values of \(n\). Need to prove that \(25\) factors out in every element
anonymous
  • anonymous
Suppose 25 divides \(16^n + 10n -1\). \(16^{n+1} + 10(n+1) -1 - (16^n + 10n -1) = 16^n.15 + 10\) Clearly, 5 divides \(16^n.15 + 10\). Dividing by 5, we have \(16^n.3+ 2\) 16^n leaves remainder 1 when divided by 5, so 5 divides \(16^n.3+ 2\) Therefore 25 divides\(16^n.15 + 10\) So, 25 also divides \(16^{n+1} + 10(n+1) -1\) . So by induction 25 divides every number in the set and is the required gcd.
rational
  • rational
Thanks!

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ikram002p
  • ikram002p
hmm do u have some other way ?
rational
  • rational
\[\begin{align}\color{blue}{16^n}+10n-1 &= \color{blue}{(1+5*3)^n}+10n-1 \\~\\ &= \color{blue}{1+5*3n + 5^2k} + 10n-1 \\~\\&= 25M\end{align}\]
rational
  • rational
recall that \((1+x)^n = 1+nx+x^2(stuff)\) from binomial thm
ikram002p
  • ikram002p
got that :)

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