# Modular arithmetic. $x \equiv 1 \pmod {m^k}$ implies $x^m \equiv 1 \pmod {m^{k+1}}$

Show that for $$k \gt 0$$ and $$m \ge 1$$, $$x \equiv 1 \pmod {m^k}$$ implies $$x^m \equiv 1 \pmod {m^{k+1}}$$

This question has already been asked in SE (Show that for $k \gt 0$ and $m \ge 1$, $x \equiv 1 \pmod {m^k}$ implies $x^m \equiv 1 \pmod {m^{k+1}}$.), but I think it´s not really answered. ( the hint given was enough but I didn’t realize it before)

I tried with Fermat´s little theorem but I get nowhere.

note: If $$a \equiv b \pmod m$$, then $$a \cdot t \equiv b \cdot t \pmod {m \cdot t}$$ with $$t \gt 0$$ (don´t know if this is useful)

Any help would be appreciated. Thanks.

• That is correctly answered (though that was just a hint). You need to look closer to the binomial expansion. For the terms $r \geq 2$ they are divisible by $m^{k+1}$, and for the term $r = 0$ it is $1$, so you really need to look at only one term... Jul 24, 2019 at 22:36
• What about the answer given to the linked question confuses you? Jul 24, 2019 at 22:37
• @HwChu that is exactly what I don´t understand about it, because as I see it, when $r=1$ the term does not contain an $m^{k+1}$ then $x^m = 1 + nm^k + qm^{k+1}$ which is not what is desired, right? Jul 24, 2019 at 22:47
• @Hw Chu now I see what you meant, thank you Jul 25, 2019 at 1:00

$$x \equiv 1 \pmod {m^k}\implies$$

There is an integer $$M$$ so that $$x = 1 + Mm^k$$.

$$x = (1 + M*m^k)$$ and $$x^m = (1+M*m^k)^m$$ and by binomial theorem will equal $$1 + Mm^{k+1} +$$ bunch of terms all times $$m$$ to power greater than $$k+1$$.

i.e.

So $$x^m = (1 + Mm^k)^m = 1 + m*M*m^k + \sum_{j=2}^m {m\choose j}M^jm^{jk}=$$

$$1 + M*m^{k+1} + m^{k+1}\sum+{j=2}^m {m\choose j}M^jm^{jk-(k+1)}$$.

(Note: If $$j \ge 2$$ then $$jk-(k+1)=(j-1)k - 1\ge k-1 \ge 0$$ so )

$$Mm^{k+1} \equiv 0 \pmod{m^{k+1}}$$ and $$m^{k+1}\sum+{j=2}^m {m\choose j}M^jm^{jk-(k+1)}\equiv 0 \pmod{m^{k+1}}$$

$$x^m \equiv 1\pmod {m^{k+1}}$$.

We have $$x^m - 1 = (x-1) (x^{m-1} + x^{m-2} + \cdots + 1)$$. Now, by hypothesis, $$m^k \mid x-1$$. On the other hand, since $$k > 0$$, we also have $$x \equiv 1 \pmod{m}$$, so $$x^{m-1} \equiv 1 \pmod{m}$$, $$x^{m-2} \equiv 1 \pmod{m}$$, ..., $$1 \equiv 1 \pmod{m}$$. Therefore, $$x^{m-1} + x^{m-2} + \cdots + 1 \equiv 1 + 1 + \cdots + 1 = m \equiv 0 \pmod{m}$$, so $$m \mid x^{m-1} + x^{m-2} + \cdots + 1$$.

Consequence of binomial expansion. If $$x=m^kn+1$$ then $$x^m=(m^kn+1)^m$$ the expansion of this has the exponent on $$m$$ in each term, except the last 2, greater than k( in fact multiples of k prior to coefficients). The last term is $$1^m$$ and the second last has $${{m}\choose {m-1}}=m$$ as a coefficient. fleabloods answer tipped me off to my mistakes.