Congruence Question: Proving m=n (mod p-1)

[StudentR]

Congruence Question !

I have a question regarding congruences, I could not find this result in the textbooks.

(note to readers: a^k means a to the power k, and = means congruent)

If we have a congruence: a^m = a^n (mod p) for a,m,n,p>0

It seems likely to deduce that m = n (mod p)

However after attempting a homework question, I discover that

a^m = a^n (mod p) implies m = n (mod p-1)

Is this result true? How does one go about to formally prove the above statement?

Thank you...

 

[shmoe]

What made you think that's true? Did you try some numerical examples?

 

[matt grime]

One would first go about thinking what the correct statement of the above should be: one raised to any power is still one.

 

[jbusc]

Consider a^m = a^n (mod p) => m = n (mod p) for m=p, n=1. Then it's obviously not true, and proof that it's not true follows immediately from the little fermat theorem.

Hint: the little fermat theorem is key to understanding the 2nd set of congruences as well