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I am attempting to find $8^{8^8}$ (which, by the way, means $8^{(8^8)}$) without any means such as computers/spreadsheets. Here's my attempt so far, and I'm pretty sure my answer is correct, but I would like a more efficient method.

First, I do the exponent: $8^8=(2^3)^8=2^{24}$, and I calculated that the last three digits are 216 by hand. I then know that $8^{(8^8)}\equiv8^{216} \pmod{1000}$, and so I have to calculate this and found that it repeats in cycles of $100$.

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Using this information, I deduce that $8^{(8^8)}\equiv8^{216}\equiv8^{200}\cdot8^{16}\equiv8^{16}\equiv2^{48}\equiv656\pmod{1000}$

Is there is a more efficient way to solve this problem than just listing out all the remainders, as I have done? I would like to keep the explanation as basic as possible, without such devices as Euler's totient function, etc.

Someone has asked me if How do I compute $a^b\,\bmod c$ by hand? is what I wanted, but no, because I want to keep it as elementary as possible, and I also don't want any tedious calculations (as I have done).

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    $\begingroup$ Using modulo of exponents is not valid. For example, $7 \equiv 2 \pmod 5$, but $2^7 = 128 \equiv 3 \pmod 5$ while $2^2 = 4 \equiv 4 \pmod 5.$ $\endgroup$
    – md2perpe
    Sep 17, 2020 at 22:39
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    $\begingroup$ I don't think this deserves a downvote. There's clearly a lot of effort. $\endgroup$
    – Shaun
    Sep 17, 2020 at 22:51
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    $\begingroup$ If you insist on using brute force cycle determination then it would be better pedagogically to choose an example with period smaller than $100$. $\endgroup$ Sep 17, 2020 at 23:00
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    $\begingroup$ It can be done easily by the first few terms of the Binomial Theorem without requiring any periodicity knowledge. If that is of interest let me know and I will post the details $\endgroup$ Sep 18, 2020 at 0:20
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    $\begingroup$ Note that you know that the answer will be divisible by $8$ so you are only really interested modulo $125$ $\endgroup$ Sep 22, 2020 at 21:03

4 Answers 4

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Without Euler's totient function, by repeated squaring, from $8^8\equiv216\bmod1000$,

we have $8^{16}\equiv656\bmod1000$, $8^{32}\equiv336\bmod1000$, $8^{64}\equiv896\bmod1000$,

and $8^{128}\equiv816\bmod 1000$, so $8^{216}\equiv8^{128}8^{64}8^{16}8^8\equiv656\bmod1000.$

And I would like to re-iterate the comment that $c^a\equiv c^b\bmod n$

does not generally follow from $a\equiv b\bmod n$.

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    $\begingroup$ That's a lot of painful mod $1000$ multiplications that are left for the reader! $\endgroup$ Sep 18, 2020 at 0:49
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Here is a way using only simple mod arithmetic and $\,\rm\color{#90f}{BT}=$ Binomial Theorem

Let $\ N := (8^{\large 8}\!-\!2)/2 \equiv -18\,\pmod{\!125}.\,$ Then by $\,\rm\color{#90f}{BT}\,$ & $\, 65^{\large 3+k}\!\equiv 0\,$ by $\,5^{\large 3}\!\mid 65^{\large 3}\,$ so

$\qquad\ \ \ \begin{align} &8^{\large 8^8-2}\! = 8^{2N}\!\!= (-1\!+\!65)^N\!\equiv -1\! +\! N\cdot 65 - \tfrac{N(N-1)}2 65^2\equiv \color{#c00}{-21}\!\!\!\pmod{\!125}\\[.2em] \Rightarrow\ &8^{\large 8^8-1}\! \equiv 8(\color{#c00}{-21})\equiv \color{#0a0}{82}\!\pmod{\!125}\\[.2em] \Rightarrow\ &8^{\large 8^8}\!\!\equiv 8(\color{#0a0}{82})\equiv \bbox[5px,border:1px solid #c00]{656}\!\!\!\pmod{\!8\cdot 125} \end{align}$

Stronger $\,8^{\large 8^8}\!\!\equiv 6656\pmod{\!8000}\,$ if we use $\!\bmod 1000$ in 2nd last congruence.

Generally the most efficient way to handle problems like this is to employ the extremely handy mDL = $\!\bmod\!\!$ Distributive Law as here to greatly decrease the modulus. Applying this law here we can pull out a factor of $\,\color{#e0f}{a = 8}\,$ from the modulus as follows
$\begin{align} ab\,\bmod\, ac \,&=\, \color{#e0f}a(b\, \bmod\, c)^{\phantom{|^|}}\!\!\!\ \ \ \ [\!\bmod\text{Distributive Law}]\\[.1em] \Longrightarrow\ 8^{\large 2+2N}\! \bmod 1000 \,&=\, \color{#e0f}8(8^{\large 1+2N}\! \qquad\,\ \bmod 125)\\ &=\, 8(8(-1\!+\!65)^N\! \bmod 125)\\ &=\, 8(8(\color{#a00}{-21})\qquad\bmod{125})\ \ \ {\rm by} \ \ {\rm \color{#90f}{BT}\ as\ above,\ and}\,\ N\equiv -18\\ &=\, 8(\color{#0a0}{82})= 656_{\phantom{|_{|_|}}} \end{align}$
Explanation: first we used mDL to factor out $\,\color{#e0f}{a=8}\,$ from the $\!\bmod\!$ to simplify the problem by reducing the modulus from $\,8\cdot 125\,$ to $\,125.\,$ So we have reduced to powering $8$ modulo $125$. By luck $\,8^{\large 2}\equiv -1\!+\!65\equiv -1\pmod{\!5}$ which we can lift up to $\!\bmod 5^{\large 3}$ by the Binomial Theorem, after writing $\,8^{\large 1+2N}\! = 8(8^2)^N\! = 8(-1\!+\!65)^N,\,$ leaving only simple mod arithmetic to finish.

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  • $\begingroup$ May I ask why $N := (8^8 - 2) / 2$ is chosen? $\endgroup$
    – Y.T.
    Sep 22, 2020 at 10:57
  • $\begingroup$ @LearningMathematics I added an explanation, using mDL to clarify the operations (this is the best way in general - compare the two approaches). Note $\,K = \color{#c00}2+\color{#0a0}2N\iff N = (K-2)/2.\,$ In OP $\,K = 8^8\,$ is our exponent. The red $\,\color{#c00}2\,$ corresponds to the two factors of $8$ we pull out, and the green $\,\color{#0a0}2\,$ comes from needing to square $\,8\,$ to get a to a simple form enabling us to easily lift powering from $\!\bmod 5\,$ up to $\bmod 5^3\,$ by the Binomial Theorem. $\endgroup$ Sep 22, 2020 at 21:01
  • $\begingroup$ Typo: "get a to a simple" should be "get to a..." $\endgroup$ Sep 22, 2020 at 21:15
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$1000=8\cdot 125$, so $\phi(1000)=4\cdot4\cdot25=400$, $8^8\mod 400 = 16$. Then $8^{16}\mod 1000=656$. So the answer is $656$.

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    $\begingroup$ @KingLogic For the last digit, and possibly for the last two digits, you can likely just look at $8^n$, see a repeating pattern, use that pattern. For the last three digits, though, that approach feels like it's too much work to be worthwhile. I don't think there are other, easier approaches either. $\endgroup$
    – Arthur
    Sep 17, 2020 at 22:47
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    $\begingroup$ The question (including first version) explicitly asks for answers without using totient, so this is not an answer. $\endgroup$ Sep 17, 2020 at 22:54
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    $\begingroup$ OP said, " I would like to keep the explanation as basic as possible, without such devices as Euler's totient function, etc." $\endgroup$ Sep 17, 2020 at 23:00
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    $\begingroup$ @JCAA You need to read more carefully - see the final paragraph of the question. Also your aswer still leaves a lot of work (exponentiation) to be performed, much of which would likely be nontrivial to beginners. $\endgroup$ Sep 17, 2020 at 23:01
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    $\begingroup$ For the record: JCAA has deleted his comments that the prior three comments were in reply to. $\endgroup$ Sep 17, 2020 at 23:42
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In a comment the OP stated

I would like an extremely elementary method, to teach to high school students with very little background in number theory.

To begin observe that

$$ 8^{8^8} = 8^{2^{24}}$$

So starting with $8$ we need to apply the squaring operation $24$ times. Moreover, at each step we only need to keep track of the last $3$ digits since the other digits in the base-$10$ expansion can't have an impact on them when squaring the numbers.

To save work we decide to find a more general squaring rule that can hopefully be reused. We'll use the formula

$ (s+t)^2 = s^2 + 2st + t^2$

on the base-$10$ expansions to get our rules.

The $\equiv$ symbol will mean that two numbers have the same last $3$ digits.

To square $8$ and get the last $3$ digits observe, where $0 \le a \le 9$, that

$\; \text{Rule08:} (a10^2 +08)^2 \equiv \quad 600a +64$

So

$\tag 1 8^2 \equiv 64$

Continuing

$\; \text{Rule64:} (a10^2 +64)^2 \equiv \quad 800a +96$
$\tag 2 {64}^2 \equiv 96$
$\; \text{Rule96:} (a10^2 +96)^2 \equiv \quad 200(a+1) + 16$
$\tag 3 {96}^2 \equiv 216$
$\; \text{Rule16:} (a10^2 +16)^2 \equiv \quad 200(a+1) + 56$
$\tag 4 {216}^2 \equiv 656$
$\; \text{Rule56:} (a10^2 +56)^2 \equiv \quad 200a+100 + 36$
$\tag 5 {656}^2 \equiv 336$
$\; \text{Rule36:} (a10^2 +36)^2 \equiv \quad 200(a+1) + 96$
$\tag 6 {336}^2 \equiv 896$

We are happy to see that we can repeatedly apply the four rules

$$ \text{Rule96} \mapsto \text{Rule16} \mapsto \text{Rule56} \mapsto \text{Rule36} \mapsto \dots $$

in a cycle; the following is all mental work.

$\tag 7 {896}^2 \equiv 816$
$\tag 8 {816}^2 \equiv 856$
$\tag 9 {856}^2 \equiv 736$
$\tag {10} {736}^2 \equiv 696$
$\tag {11} {696}^2 \equiv 416$
$\tag {12} {416}^2 \equiv 056$
$\tag {13} {056}^2 \equiv 136$
$\tag {14} {136}^2 \equiv 496$
$\tag {15} {496}^2 \equiv 016$
$\tag {16} {016}^2 \equiv 256$
$\tag {17} {256}^2 \equiv 536$
$\tag {18} {536}^2 \equiv 296$
$\tag {19} {296}^2 \equiv 616$
$\tag {20} {616}^2 \equiv 456$
$\tag {21} {456}^2 \equiv 936$
$\tag {22} {936}^2 \equiv 096$
$\tag {23} {096}^2 \equiv 216$
$\tag {24} {216}^2 \equiv 656$

Observe that $8^{2^{22}} \equiv 8^{2^2} \equiv 096$ so the squaring sequence was just starting to repeat - the last two rule based calculations were not necessary.

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