1966 /www/nrich/html/content/98/08/15plus1/ 1 2011-02-01T00:00:01 <?xml version="1.0" encoding="UTF-8"?> <mdoxml version="1.0"><br></br> Find the values of $n$ for which $1^n + 8^n - 3^n - 6^n$ is divisible by 6. <br></br> <br></br></mdoxml> <?xml version="1.0" encoding="UTF-8"?> <mdoxml version="1.0"><br></br> <span class="editorial">Murat from Turkey sent the following solution.</span> <p>If $g(n)=(1+8^n-3^n)$ is divisible by 6, then $(1+8^n-3^n-6^n)$ is also divisble by 6. It can be verified that</p> <p>$$g(n)-73g(n-2)+576g(n-4)=504$$</p> <p>for all positive integers $n&amp;gt; 3$. Since $g(1)$ and $g(3)$ are divisible by 6, it follows that $g(5)$ is also. By induction, it can be shown that for all odd $n$, $g(n)$ is divisible by 6. Since $g(2)$ and $g(4)$ are not divisible by 6, this is not the case for even $n$.</p> <p>An alternative approach is to use the facts that powers of odd numbers are always odd; powers of even numbers are always even; also the difference of two odd numbers is even. Hence $N=1^n+8^n-3^n-6^n$ is even (odd + even - odd - even).</p> <p>It remains to decide whether or not $N$ is divisible by 3.</p> <p>$N \equiv 1 + (-1)^n - 0 - 0$ (mod 3)</p> <p>This shows that $N \equiv 0$ (mod 3) if $n$ is odd and hence $N$ will be divisible by 6 for all odd values of $n$. However, $N \equiv 2$ (mod 3) if $n$ is even and so $N$ cannot be divisible by 6 for even values of $n$.</p></mdoxml> <?xml version="1.0" encoding="UTF-8"?> <mdoxml version="1.0"><br></br> <h3><span style="font-weight: bold;">Why do</span> <a style="font-weight: bold;" href="http://nrich.maths.org/public/viewer.php?obj_id=1966">this problem</a><span style="font-weight: bold;">?</span></h3> As a simple exercise using modulus arithmetic where learners can make and test their own conjectures. By trying small values of n learners may make their own conjectures about when the expression is divisible by 6 and when it is not.<br></br> <br></br> <h3><span style="font-weight: bold;">Key question</span></h3> How can we write the expression as an equivalent expression modulo 3?<br></br> <br></br></mdoxml> <?xml version="1.0" encoding="UTF-8"?> <mdoxml version="1.0"><br></br> Consider separately when $N = 1^n + 8^n - 3^n - 6^n$ is divisible by $2$ and when it is divisible by $3$. <br></br></mdoxml> 2 3 0 0 0 1 1 Find the values of n for which 1^n + 8^n - 3^n - 6^n is divisible by 6. Numbers and the Number System Divisibility Using, Applying and Reasoning about Mathematics Mathematical induction Numbers and the Number System Modulus arithmetic Admin Short problems Secondary Mapping Document MD Factors, multiples and primes