Find $x^{98}$ congruent to $7$ (mod $18$)

Question

Find $x$, such that $$x^{98} \equiv 7 \mod 18$$ holds.
Reminder: Euler's totient function is given by $$\phi(n) = n\prod_{p_i|n}\left(1-\frac1{p_i}\right).$$

I actually don't know, how to even approach this. What is the common way, to approach this type of question? How do i start solving this? I tried simplifying to this:

\begin{align} x^{98} &\equiv 7 \mod 2 \Rightarrow x^{98} ≡ 1 \mod 2\\ x^{98} &\equiv 7 \mod 9 \end{align} but i'm not sure if this is the right approach.

Any help would be appreciated.

Answer 1

The idea is to resort to Euler's Theorem. You want to solve the equation $$x^{98}\equiv 7\,(\text{mod } 18).$$ This, in particular, would imply that $x^{98}=7 + 18\cdot n$ for some $n\in\mathbb{N}$, hence $x$ has to be coprime with $18$ (if $p$ divides $x$ and $18$, then it divides $x^{98}$ and $18$ and so $7$ as well, which is impossible). Thus you are in the hypothesis of Euler's Theorem: $x$ and $18$ are coprime and so $$x^{\varphi(18)}\equiv 1\,(\text{mod } 18).$$ By repeatedly using this relation, you should be able to simplify your equation to something more handy.

Answer 2

If $x$ is relatively prime to $18$ then $x^n$ has to be relatively prime to $18$ too. There are only six residues modulo $18$ that do this, the rest are multiples of $2$ or $3$. So $x^0=1, x^1=x, x^2, ...$ cycle among these six residues and you have $x^6\equiv x^0=1$.

Then $x^{98}\equiv x^{92}\equiv x^{86}\equiv ...$ and you get all the way down to $x^2$ where $2$ is the remainder when you divide $98$ by $6$. And solving $x^2\equiv 7 \bmod 18$ will be easy.

Answer 3

The form of Euler's totient function $\phi$ given there is not very practical, and I notice that you didn't give the value for $\phi(18)$. Here's it's easier to use that for $p$ prime, $\phi(p^k) = p^{k-1}(p-1)$ and that $\phi$ is multiplicative between coprime numbers.

So $\phi(18) = \phi(3^2)\phi(2) = 3^1(3-1)(2-1) = 6$.

Thus any number $a$ coprime to $18$ has $a^6\equiv 1 \bmod 18$ (Euler's theorem).

Since $x^{98}\equiv 7$ is coprime to $18$, $x$ is also. Applying $x^6\equiv 1$ we can see that $x^{96}\equiv 1\bmod 18 $ and thus $x^{98}\equiv x^2\bmod 18$. By inspection, since $7+18=25$, we can see that $x\equiv 5$ is an answer to $x^2\equiv 7 \bmod 18$ and thus $x\equiv -5\equiv 13 \bmod 18$ is also an answer.

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A couple of notes:

• Because we ended up looking for $x^2\equiv 7 \bmod 18$, we needed that $7$ is a quadratic residue $\bmod 18$, which is only true of half the values coprime to $18$. If the question had $5$ in place of $7$, there would be no answer.
• If we were looking at a modulus divisible by two odd primes, or by an odd prime and $4$, there would be more than two answers (for any soluble system). For example $x^2\equiv 13 \bmod 36$ has $x\equiv \{7, 11,25,29\}\bmod 36$.

Answer 4

Thank you for the answers,

I hope I got it right so I solved this question for example:

Find the last digit of $7^{2013}$

so obviously

$7^{2013} ≡ x(mod 10)$

so my thought process was the same, i know that $gcd(7,10)=1$

$φ(10)=φ(5-1)φ(2-1)=4$

then $(7^4)^{503}*7≡x(mod10)$

finally: $$7≡x(mod10)$$

so the final answer is 7, I hope I got it right...