Martha's Expression and Hypothesis

Level 1

Martha investigates the expression $9^n-8^n$ . She substitutes 1, 2, 4, 6, and 12 for $n$ and here are her results:

$9^1-8^1=1$ , which shares no common factors with 18 other than 1.
$9^2-8^2=17$ , which again shares no common factors with 18 other than 1.
$9^4-8^4=2465$ , which yet again shares no common factors with 18 other than 1.
$9^6-8^6=269297$ , which still shares no common factors with 18 other than 1.
$9^{12}-8^{12}=213710059745$ , which unsurprisingly still shares no common factors with 18 other than 1.

She makes a hypothesis that the expression she was investigating will never share any common factors with 18 other than 1 for any positive number $n$ .

Is her hypothesis right?

No, it depends on the value of

n

Yes, her hypothesis is true for any positive integer

n

1 solution

X X
Sep 8, 2018

$9^{n+1}-8^{n+1}=9^n+9^{n-1}8^1+9{n-2}8^2+...+9^28{n-2}+9^18^{n-1}+8^n$

$=(9^n+9^{n-1}8^1+9{n-2}8^2+...+9^28{n-2}+9^18^{n-1})+8^n$ (which is obviously not a multiple of 3.)

$=9^n+(9^{n-1}8^1+9{n-2}8^2+...+9^28{n-2}+9^18^{n-1}+8^n)$ (which is obviously not a multiple of 2.)

Hence, every positive $n$ will make the expression and 18 coprime.

Martha's Expression and Hypothesis

1 solution

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