If $\operatorname{ord}_ma=10$, find $\operatorname{ord}_ma^6$

Question

If $\operatorname{ord}_ma=10$, find $\operatorname{ord}_ma^6$. Justify you're answer.

I'm not sure what I should say for my answer to be justified. However, I expect $\operatorname{ord}_ma^6=5$, because $5\cdot 6=30$ and $10\mid 30$.

Like I said, no idea if this is how to go about solving this problem.

I expect that this, together with "No lower positive multiple of $6$ is divisible by $10$" should do the trick. — Arthur, Nov 04 '12 at 21:09

score 1 · Answer 1 · edited Apr 13 '17 at 12:20

Hint $\ $ If $\rm\ ord\ a\, = 10\:$ then $\rm\:a^{6n}\equiv 1\iff 10\mid 6n\iff 5\mid 3n\iff 5\mid n.$

Remark $\ $ Above we invoked: $\ $ if $\rm\:ord\ a\, = k\:$ then $\rm\:a^n\equiv 1\iff k\mid n.\:$ This can be viewed much more conceptually: the set $\cal O$ of integers $\rm\,n\,$ with $\rm\:a^n\equiv 1\:$ is closed under subtraction, hence, by a fundamental theorem, every element of $\cal O$ is a multiple of the least positive element of $\cal O,\,$ which in our case is called the order.

score 1 · Accepted Answer · answered Nov 05 '12 at 12:20

1

If $$ord_ma=d, ord_m(a^k)=\frac{d}{(d,k)}$$ (Proof @Page#95)

Here $$d=10,k=6\implies ord_m(a^6)=\frac{10}{(10,6)}=\frac{10}2=5$$

answered Nov 05 '12 at 12:20

lab bhattacharjee

274,582

The linked proof is also on our site, e.g. see this simpler one-line proof. – Bill Dubuque Nov 05 '12 at 20:09

If $\operatorname{ord}_ma=10$, find $\operatorname{ord}_ma^6$

2 Answers2

Linked