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2 Let p be the smallest prime which is bigger than the last two digits of your student ID number (10 points) How many primitive roots are there modulo p, 2pl0 and 3...

Question

2 Let p be the smallest prime which is bigger than the last two digits of your student ID number (10 points) How many primitive roots are there modulo p, 2pl0 and 3pl0? (10 points) Find primitive root g modulo p- (10 points) What is the order of g? modulo p (10 points) Use the primitive root g to express all the quadratic residues modulo p as powers of g (10 points) Is 5 a quadratic residue modulo p? Justify your answer_

2 Let p be the smallest prime which is bigger than the last two digits of your student ID number (10 points) How many primitive roots are there modulo p, 2pl0 and 3pl0? (10 points) Find primitive root g modulo p- (10 points) What is the order of g? modulo p (10 points) Use the primitive root g to express all the quadratic residues modulo p as powers of g (10 points) Is 5 a quadratic residue modulo p? Justify your answer_



Answers

How many different ways can I write the number 10 as the sum of some prime numbers if order does not matter.

Um so you X squared plus X minus p is equal to zero where P is prime. Um, Now, if PS prime, that means that the only factors of P could be a one and p so are possible. Rational roots rational roots need to be plus or minus one or plus or minus p. So if we did synthetic division on this, um, so 11 negative p. I think you'll see that one probably will not get you a remainder of zero. Uh, let's see if we have one here one times one is one. Um, and then adding straight down is to two times want to give me too. And so if we get a remainder of zero, we're adding straight down. Um, native P plus two must equal zero. So as we subtract two over and then divide by name one, this works if P is the prime number two. Okay, so P equals two is a possibility. Let's try for negative one. Now, I just seen if this will give me a different answer. So bring the one down and multiply by native one, um, and then adding sure dining at 00 times anything is zero, and we're supposed to get a remainder of zero. But if we add Ning of people zero well, zero is not a prime number. So that does not work, so I can cross that off. Let's see if we get any other answers. If if we're trying to divide by P 11 p. So bring the one down one times p S p one plus p one plus p times p is P plus p squared. We need to get a remainder of zero. So as we add straight down and get two p plus p squared, Um, well, it's a factor out of P two plus p. So this equation works if P equals zero or P equals negative, too. Okay, well, that's kind of the same thing we had appear. Um, let's just let's see if I can squeeze us working right here. Let's say that's negative. P will be negative and this be negative, while negative times a negative will make it plus right there. So P plus native PB zero. So that would be just setting zero equals p squared so P would equal zero. Okay, so that didn't work either. So the only answer that works is P equals two and hopefully my work right here kind of explains it And I cant went off a tangent to show why no other answer works to is the only one

We have a question in this way to use calculators to find and to a plus one from 1 to 10. And uh we have to find the smallest natural number and for which end plus. And this one is not a crime. Basically what we have to do is to find and Victoria Plus one so then And equal to one. and Victoria Plus one will be equal to one factorial plus one. That is to when an equal to two, it will be two factorial plus one, that is three When and his three. This will be three victoria plus one. That is six plus one which is seven when an equal to four. This will be four victoria plus one, That is 24 plus one Which is 25 When an equal to five. This is five pictorial plus one. So this will be 24.120 Plus one. That is 121 One and equal to six is a six pictorial plus one. That is 720 plus one, And often unequal to seven, it is seven Victoria Plus one. Equal to 5040 plus 15041. But instead of doing this. Instead of doing this, we have to find the smallest natural number for me. This is not a crime. So we can see that this is a prime number. This is a prime number. This is a prime number but This is not a prime number. This is divisible by five, sure, smallest natural number for which. and factory Plus one is not a primary is for They should be the answer. Thank you.

So in order for us to find the remainder of this, the best way to go about this is to look at this in terms of a module. So the modular we want to look at is module 15th. We want to look at seven. The 1020 marred 15. Now remember what this means is we just look at the remainder of that number at any point. So what I want to do Is first rewrite seven and we'll get some powers of seven. So let's do seven squared. So this is 49. So if we were to look at seven squared or just 49 Maude 15, remember this just tells us what the remainder of that would be. So if we were divide 15 x 49, 49 x 15, that would be Just a remainder of four because 15 would go in three times 45. subtract that office. It would be four would be equal to four mod 15. That means if we were to rewrite seven as seven squared, raised to the 510th power, This is 49. But we just said 49 model 15 is still equal to four. So this would be four Raised to the 5, 10 Modular 15. So now let's look at some powers of four. Well we know four squared is 16 and all 16 Maude 15 is going to be equal to want. So if we go about doing that, it would be four squared raised to the 255th I believe 1 15. So that's 16 and then we just said 16 mile 15 is one would be one raised to the 255 and well, one Or one race to the 255 powers is going to be one. So that's one maud 15. So we just showed was that 7 to the 10:20 um, mod 15 is equal to one mod 15, which implies That what we do 720 divided by 15 and we look at the remainder, this is just equal to one, So our remainder is one.

So this question we're asked, how many different ways Can we write the number 10 as the sum of prime numbers if order, does not matter remember what a prime number is. A prime number is a number having only two factors one and itself. I think there are two ways That you can write the number 10 as the sum of prime numbers. If water does not matter, I think you can write it as three plus seven, three plus seven equals 10. Three is prime Because the only two factors of three Are one and 3, seven is also prime Because the only two factors of seven are 1 and seven. The other way I could write 10 as the sum of prime numbers is five and 5, five Plus 5 is 10. five is a prime number because It only has two factors 1 and itself And again five is a prime number because it only has two factors one and itself. These are the only two ways that you can write the number 10 as the sum of prime numbers. Hopefully this makes sense. Have a great rest of your day.


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