Divisibility of an expression by 323

Question

Determine the number of integers $n$ with $1\leq n\leq 2017$ such that $323$ divides the expression $20^n + 16^n - 3^n - 1$.

So first, the expression is an even number (not sure what that does).

Then I try $323=20\times 16 + 3\times 1$ and so I get a $X = \frac{(20^n + 16^n)-(3^n+1)}{20\times 16} +3\times 1$ and then I don't know how to factor something out or something.

And so I try pressing some values in calculator and I see all even numbers for $n$ work so I get 1008, which is correct but how do I do this like logically?

$323=17\times19$ surely? – Angina Seng May 21 '18 at 15:45 — Angina Seng, May 21 '18 at 15:45
oh no i forgot that, oh no ok i tot it was prime – shgdh fgxbcv May 21 '18 at 15:46 — shgdh fgxbcv, May 21 '18 at 15:46
I Find it hard to believe 1008 cases work !? – mick May 21 '18 at 15:50 — mick, May 21 '18 at 15:50

score 0 · Accepted Answer · answered May 21 '18 at 15:53

0

We have $$20^n\equiv 3^n\mod 17$$ and $$16^n\equiv (-1)^n\mod 17$$ so we have $$20^n+16^n-3^n-1\equiv 3^n+(-1)^n-3^n-1\mod 17$$ and $$20^n\equiv 1\mod 19$$ and $$16^n\equiv (-3)^n\mod 19$$ so we get $$1+(-3)^n-3^n-1\mod 19$$ Can you finish?

answered May 21 '18 at 15:53

Dr. Sonnhard Graubner

95,283

Very nice answer Prof! ٩(˘◡˘)۶ – – mick May 21 '18 at 16:20
1

The main thing is that your problem was solved, have a nice evening! – Dr. Sonnhard Graubner May 21 '18 at 16:22
Not MY problem lol. Although I was surprised at first that it divides So often. :) Im not the Op but Thanks anyway lol – mick May 21 '18 at 16:24

user · Answer 2 · 2018-05-21T16:05:49.027

0

HINT

Since $323=17\times 19$ by CRT we need to solve

$20^n + 16^n - 3^n -1\equiv 0 \iff 3^n + (-1)^n - 3^n -1\equiv 0 \\\iff (-1)^n+1\equiv 0 \mod{17}$
$20^n + 16^n - 3^n -1\equiv 0 \iff 1 + (-3)^n - 3^n -1\equiv 0\\\iff-2\cdot3^n\equiv0 \mod{19}$

edited May 21 '18 at 16:05

answered May 21 '18 at 15:55

user

154,566

Ok thank you but i think it should be (-3)^n -3^n in the end line? – shgdh fgxbcv May 21 '18 at 16:02
@shgdhfgxbcv ops...yes of course! fixed – user May 21 '18 at 16:06

Divisibility of an expression by 323

2 Answers2