Computing $75^{75^{75}}$ modulo $32$

Question

How to compute $75^{75^{75}}$ modulo $32$?

I tried:

It's $\mathrm{gcd}(75,32)=1$, so with Euler's phi function I get

$\varphi(32)=32(1-\frac{1}{2})=16$.

Then it's $75^{16}\equiv1 \ (\mathrm{mod} \ 32)$.

So I computed $75^{75}$ modulo $16$:

It's $\varphi(16)=16(1-\frac{1}{2})=8$

So $75^{75}\equiv 1 \ (\mathrm{mod} \ 8)$ and $75^{75^{75}} \equiv 1 \ (\mathrm{mod} \ 8)$

Then it's $75^{75^{75}}\equiv 75^1 \equiv 75 \ (\mathrm{mod} \ 32)$.

I'm not sure if this is correct. Is there something wrong in this calculation or can it be done like this?

The middle part is wrong because $75^{75}\not\equiv 1 \bmod 16$. — Dietrich Burde, May 15 '20 at 16:36
Right ideas in there. But isn't $75\equiv3\pmod8$. Therefore $75^{75}\equiv75^3\pmod{16}$ et cetera. — Jyrki Lahtonen, May 15 '20 at 16:37
I think all the variants of this type of problems are handled in a mother duplicate. Some users think that iterated powers are not handled adequately there, and oppose using it as a duplicate. I think it should. Do check the list of questions linked to it! Plenty of material there. — Jyrki Lahtonen, May 15 '20 at 16:44
And "some users think" that dupe links should be specific and high quality - not wild goose chases such as "maybe one of the motley answers here can be applied in some way to solve your problem". — Bill Dubuque, May 15 '20 at 17:00

score 1 · Accepted Answer · answered May 15 '20 at 17:03

1

$$75^{75 } \equiv 75^{72} 75^3 \equiv (75^{8})^{9}11^3 \equiv 3 \mod 16$$

so we can say $$ {75^{75}}^{75} \equiv 75^3 \equiv 11^3 \equiv 19 \mod 32$$

answered May 15 '20 at 17:03

Ziad Fakhoury

How to see fast that $75^3 \equiv 11^3 \equiv 19 \ (\mathrm{mod} \ 32)$? – Tartulop May 16 '20 at 11:44
Well $75 = 2(32) + 11$ a simple application of the euclidean algorithm. As for $11^3$ I just computed it to be $1331$ and I noticed at first $1331 = 1024 + 307 = 2^{10} + 307 = 32^2 + 307$. Then just euclidean algorithm on $307$ o get $307 = 9(32) + 19 $ – Ziad Fakhoury May 17 '20 at 20:06

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