Find $\,n\bmod 42\,$ given $\,7\mid 3^n+4n+1$

Question

I have been working on a problem relating to remainders: "The number $3^n+4n+1$ is divisible by $7$. Find the remainder of n when it is divided by $42$." I have been unable to find a strategic way to solve this problem and so far, I have only found the remainders, $4$, $8$, $13$, $21$, $23$, and $24$. Can anyone confirm whether this is correct?

Answer 1

Key Idea $ $ We eliminate nasty exponential dependence on $\,n\,$ in $\,3^{\large n}\,$ by noting - by Fermat - that it's periodic $\!\bmod 7\!:\ 3^{\large 6}\equiv 1\,\Rightarrow\, 3^{\large n}\! =3^{\Large i\large +6k}\equiv \color{90f}{3^{\Large i}}\,$ by mod exponent reduction. Thus we get a simpler linear congruence by replacing the function $\ g(n) = \color{#90f}{3^{\large n}}\,$ by constants $\,\color{90f}{3^{\Large i}},\ i = 0,1,\ldots,5$.

Doing so $\ n = \color{}i\!+\!6\,\color{#c00}k\,\Rightarrow$ $\! \bmod\color{#c00} 7\!:\,\ \overbrace{0\equiv f(i\!+\!6k)\equiv \color{90f}{3^{\Large\color{}i}}\!+\! 4(i\!+\!6k)\!+\!1}^{\Large{\ \ \ \ 7\ \, \mid\, \ f(n)\ \ =\ \ \color{#90f}{3^{\LARGE n}}\ \ +\ \ 4\,n\ \ +\ \ 1}} \equiv f(i)\!+\!3k\!$ $\iff\!$ $ 3k\equiv -f(i)\!\equiv\!6f(i)\!\!\iff\!\!$ $\color{#c00}{k \equiv 2f(i)}^{\phantom{|^I}}\!\!\!,\, $ so $\, n^{\phantom{|^I}}\!\!\! = i\!+\!6\color{#c00}k = i\!+\! 6(\color{#c00}{2f(i)\!+\!7}j)\equiv \bbox[5px,border:1px solid #0a0]{i\!+\!12f(i)}^{\phantom |}\!\pmod{\!42},\, $ e.g. $\,i\!=\!0^{\phantom{|^{|^|}}}\!\!\!\Rightarrow n\equiv 0\!+\!12f(0)\equiv\bbox[5px,border:1px solid #0a0]{24},\,$ $\ i\!=\!1\Rightarrow n^{\phantom{|^I}}\!\!\!\!\equiv 1\!+\!12f(1)\equiv\bbox[5px,border:1px solid #0a0]{13},\,$ $\, i\!=\! 2,3,4,5\Rightarrow \bbox[5px,border:1px solid #0a0]{n\equiv \ldots\phantom{i}\!\!\!}$

Remark $ $ The same idea works generally. If $\,g(n)\,$ has period $\,\ell,\,$ so $\ g(i+ \ell k) = g(i),\,$ then we can solve $\,f(n,g(n)) \equiv a\pmod{\!m}\,$ by using this periodicity to eliminate the function $\,g(n)\,$ as above.

Doing so $\ n = i+\ell\, k\,\Rightarrow\bmod m\!:\ a\equiv f(n,g(n))\equiv f(n,g(i))\iff \ldots\, n\equiv r_i$

Combining $\,\ \begin{align} &n\equiv\, i\pmod{\!\ell}\\ &n\equiv r_i\!\!\!\pmod{\!m}\\\end{align}\!\! \overset{\large \rm CRT\!\!}\iff n\equiv s_i \pmod{\!\ell m},\ $ with $\,s_i\,$ computable by CRT,

assuming $\,\ell,\,m\,$ are coprime. In our solution above we did the CRT step in parallel by substituting $\,n = i + \ell k\,$ not only into $\,g(n)\,$ but everywhere for $\,n\,$ in $\,f,\,$ then solve that for $\,k \equiv k_i\pmod{\!m},\,$ so $\, n = i +\ell (k_i + mj)\equiv i + \ell k_i \pmod{\!\ell m}$.

Answer 2

Strategy:

$4n + 1 \equiv k \pmod 7$ will have a solution $n_k$ determined by $n \mod 7$. And for $k = 0,... 7$ there will be $7$ solutions $n_0, ... n_6$.

By Fermats little Theorem, $3^6\equiv 1 \pmod 7$ for $3^n \equiv j\pmod 7$ will have a solution $m_j$ determined by $n \pmod 6$. If we assume $3$ is a primitive root (an assumption for now but one easily determined by simple calcultions) for $j= 1,2,3,...6$ there will be six solutions: $m_1,...,m_6$

So to have $3^n + 4n + 1\equiv k - k \equiv 0 \pmod 7$ is a matter of finding the six $n_k, m_{7-k}$ pairs and solving $n \equiv n_k \pmod 7$ and $n_{7-k} \pmod 6$.

By the Chinese Remainder Theorem, for each $n_k, m_{7-k}$ pair there will precisely one solution $\mod 6*7$.

======= Full Answer below=====

The strategic way I'd consider would figure $3^n$ has a cyclic $\mod 7$ and figure what that is. Presumable $3$ is a primitive root and the cycle is and $3^n \equiv 1,3,2,6,4,5 \pmod 7$ if $n \equiv 0,1,2,3,4,5,6 \pmod 6$

Meanwhile $4n + 1 \equiv 1,5,2,6,3,0,4 \pmod 7$ when $n \equiv 0,1,2,3,4,5,6,7$

We need $3^n + 4n + 1 \equiv 0 \pmod 7$ so

if $n \equiv 0 \pmod 7$ then $4n+1\equiv 1$ and we $3^n\equiv -1$ so we would need $n \equiv 0 \pmod 7$ and $n\equiv 3\pmod 6$.

Similarly if $n\equiv 1,2,3,4,5,6 \pmod 7$ we have $4n+1 \equiv 5,2,6,3,0,4 \pmod 7$ and we need $3^n \equiv 2,5,1,4,0,3 \pmod 7$ so we need $n\equiv 2,5,0, 4, impossible,1 \pmod 6$.

Now by CRT th each of these pairs of options has a unique solution:

1) If $n\equiv 0 \pmod 7;n\equiv 3 \pmod 6$ then $n \equiv 21 \pmod {42}$.

2) If $n\equiv 1 \pmod 7;n\equiv 2 \pmod 6$ then $n \equiv 8 \pmod {42}$.

3)If $n\equiv 2 \pmod 7;n\equiv 5 \pmod 6$ then $n \equiv 23 \pmod {42}$.

4)If $n\equiv 3 \pmod 7;n\equiv 0 \pmod 6$ then $n \equiv 24 \pmod {42}$.

5) If $n \equiv 4 \pmod 7; n\equiv 4\pmod6$ then $n \equiv 4 \pmod {42}$

6) If $n \equiv 6 \pmod 7; n\equiv 1\pmod 6$ then $n \equiv 13 \pmod {42}$.

Answer 3

Let's say that $n=42a+b$, with $a \in Z$ and $b \in \{0,1,2,3,...,41\}$, since $n\in Z$.

Substituing we get $ 3^n+4n+1=0 (mod 7) \Rightarrow 3^{42a+b}+4(42a+b)+1=0 (mod 7)$

$ \Rightarrow 3^b.(3^{42})^a+7.(24a)+4b+1=0 (mod 7) \Rightarrow 3^b.(3^{42})^a+4b+1=0 (mod 7)$

Note that $3^{42}=({3^2})^{21}=9^{21}=(7+2)^{21}=2^{21}=64=1 (mod 7)$.

So $3^b.(1)^a+4b+1=0 (mod 7) \Rightarrow 3^b+(7-3)b+1=0 (mod 7) \Rightarrow 3^b-3b+1=0 (mod 7)$

Now we only have to test all values of $b \in \{0,1,2,3,...,41\}$ which $ 3^b-3b+1$ is multiple of 7, remaining 4, 8, 13, 21, 23 and 24.