5

If p is prime then aP = a (mod p) for all a ez_(6 points) Useto show that a and a5 have the same units digit for any integer...

Question

If p is prime then aP = a (mod p) for all a ez_(6 points) Useto show that a and a5 have the same units digit for any integer

If p is prime then aP = a (mod p) for all a ez_ (6 points) Use to show that a and a5 have the same units digit for any integer



Answers

Use Exercise 62 to show that if $p$ is an odd prime and $a$ and $b$ are integers not divisible by $p$ , then
$$
\left(\frac{a b}{p}\right)=\left(\frac{a}{p}\right)\left(\frac{b}{p}\right)
$$

So in this question, we want to show that if he is a crime, then such that piece on divisible by a then eight p months to is an inverse or a more. Hey, so despite definition off for my little theory, we have a two piece must buy. One is equivalent to one model p of assistance. Just spirits. That's all. L R T T O E. Yeah. Famous. It was here so we can decompose eight p months by one as being eight Pete miles bite you times by a and this is equivalent to one. Well, he and we already have a results because a times by eight he minds by choose gives you one month p so ends on therefore the inverse of a more he is this he wants to

Okay, so what this question? I want to show that X squared is equivalent to a more he that has no solution. Courts that to include your solutions. So he has no solutions. If a is not in the set off on critical issues Quadratic residues. Oh, he Now we then assume that a is Emiko Jared was juiced off peak. So then therefore there exists a solution equipment to a more he Now we also assume that why is a mother submission to this? So then because they're equal wasn't just said X squared is equipments to y squared Well, he mom so x squared. My wife sport is acquitted. That's 20 more he so therefore he's a divider off x squared minus y square. But we can factor exploit my wife's quit It's well hidden devices off X miles by Why x plus y So since peace device out these you have he is devising off x marks for why? Well, he is devising off X plus y therefore X minus y y month. Hey, is it to czar or he is equal base Plus for why mother he is your accuser. To combine these two results we have X is equivalent to Why? Well, he oh X is equivalent to negative one monkey. So hence we have two solutions. Hence we either have no solution. When a lot in the cortex residues Well, we have two solutions being positive. One negative ones.

So we know that is hey is equivalent to pee. What p then either both a and be are court drive it present to or none residue off people in each of these case, um, first case, second case. In each of these case, these cases we would have a overpay. It was to be over peace.

So we know that if a is equivalent to be want P so either both A and B are or drought it residue U two's or none residue of P. Remember, as you do about residue or residue of people, so in either of the cases, we would have a overpay in closed to be overpay.


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