## Odd Power Divisibility

#### Problem

Prove that $6^n + 8^n$ is divisible by 7 iff $n$ is odd.

#### Solution

We shall prove this in two different ways. The first proof, which is the simplest, makes use of congruences and the second proof makes use of the binomial expansion .

First Proof

As $6 \equiv -1 \mod 7$ and $8 \equiv 1 \mod 7$ it follows that $6^n \equiv (-1)^n$ and $8^n \equiv 1$.

$\therefore S = 6^n + 8^n \equiv (-1)^n + 1 \mod 7$.

If $n$ is even then $(-1)^n = 1 \implies S \equiv 2 \mod 7$.
If $n$ is odd then $(-1)^n = -1 \implies S \equiv 0 \mod 7$.

Thus S is divisible by 7 iff $n$ is odd.

Second Proof

Let $x = 6^n = (7 - 1)^n$ and $y = 8^n = (7 + 1)^n$.

\begin{align}\therefore y &= 7^n + \dbinom{n}{n-1}7^{n-1} + \dbinom{n}{n-2}7^{n-2} + ... \dbinom{n}{2}7^2 + \dbinom{n}{1}7 &+ 1\\x &= 7^n - \dbinom{n}{n-1}7^{n-1} + \dbinom{n}{n-2}7^{n-2} - ... &+ &\pm1\\\therefore x+y &= 2 \times 7^n + 2 \dbinom{n}{n-2}7^{n-2} + 2 \dbinom{n}{n-4}7^{n-4} + ...\end{align}

We note that the last term of the series for $x$ will be $-1$ if $n$ is odd and $+1$ if $n$ is even. Therefore the series for $x + y$ will end $2 \dbinom{n}{1}7$ if $n$ is odd and 2 if $n$ is even. In other words, all the terms will divide by 7 except the last term when $n$ is odd.

Hence $x + y \equiv 0 \mod 7$ iff $n$ is odd.

What can you deduce about $x + y \mod 49$?
Investigate the remainder when $(a - 1)^n + (a + 1)^n$ is divided by $a$ or $a^2$.

Problem ID: 279 (13 May 2006)     Difficulty: 3 Star

Only Show Problem
{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 1, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9954632520675659, "perplexity": 372.7423502826619}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2014-15/segments/1398223211700.16/warc/CC-MAIN-20140423032011-00106-ip-10-147-4-33.ec2.internal.warc.gz"}