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Find the Question 1 number of Integers 13 which are prlmitive roots modulo...

Question

Find the Question 1 number of Integers 13 which are prlmitive roots modulo

Find the Question 1 number of Integers 1 3 which are prlmitive roots modulo



Answers

Find the remainder when $x^{3}-3$ is divided by $x+1.$

Okay, So this question we want to fuse the master. Little steering to find 7 to 1 to one more sexy. So what does Fermat little steering safe? It saves that eight to the He might buy one. Um, is equivalent to one more. He? Yeah. Oh, was he is crime and a isn't visit buckets. Opie's a he doesn't, but he does. So first we know that. 13. The client That's pretty sweet. So formatted perfume in relation to this is that seven to the P Morris by once or 13 miles by one that gives you 12 is equivalent to one more 30. So what we want to do, then, is we want to factor out as many 7 to 12 as we can from this number. Right. So we have seven to the 1 to 1. He's even to 7 to 1 20 plus one. This is equal to seven. Has a 7 to 1 20 just to seven times by seven to the £4. 10. This is just seven tons by 7 to 12 to 10. So when we take month 13 off this more 13 by firm ass little serum, this the thing in the bracket becomes one. So 1 to 10 just give you one. So this is just equal or equivalent to seven. Mother, let's see.

So we want to find which into just leave remained off one divided by two liver. Leave a remainder off. One was divided by three. So we have to write this in terms of models. So one leaving a remainder of one divided by two means that X will take more, too. This must give you one and x month. Three must positive one. So rewriting this in terms off more familiar terms we have X is equivalent to one more, too, and X X is equivalent to one mo three. So then we have a system off equations or congruence relations, and we want to solve this so we can either solve this by China's remain a serum. Oh, we can just use back substitution, which I'll just use mine. So expand this out. You have X is equal to one class by two. Hey, well, a something to jump something to this interview we have one plus by two is equivalent to one one more. Three. Subtract one over to a is equivalent to zero more. Three. Andi invested. Two month three is just too is equivalent to two time by zero more three, which is just equivalent to zero more. Three. And we write this as three time Bobby for be some instant. Yeah. Substitute this back into here. We have X is equal X is equal to one plus by two tons of eight, which is it to three. But then this is equal to one despite six feet. And we can convert this into mods. Being X is equivalent to one more six.

So we know that the is a prom. It'd route of 17. We now have a equals. One would be equals two, See to the power of zero blood 17. This can't be right. Um, rewrite as lock three a or locks through you one. This equals 20 Again. A equals channel is equal to three to the power off. 14 Mott 17. We can rewrite this us. See block off the two equals two. 14. A host a is saving us equals toothy to the power off. One month. 17 So I can write Lock three of three is most to one. So a equals 43 to the power of four. But it was not a to the power for three to the power of 12. Yep. 17. This would be equals that love. Three for equals to 12. Um, so all of this I'm just going to write down a general rule, so it's easier for us to follow. So is blocked. Three of a so 80 who? Sly a was life. This equals two feet to the power of five. March 17 rewritten as love three. Five e close to five. So a equal six equal six equals to t to the power of 16. Mont. 17. We written US lock see 60 lofty A, which is locks three of six equals 2 16 A equal seven is eat most to the to the power of 11. But 17 we would then have long three of seven is equals. To align a equals eight IHS equal suit the to the power of 10 ma 17. We can rewrite those as lofty off eight equals 2 10 A close nine is equal to three to the power of to Mott. 17. So law three of nine is equals to two. A equals 2 10 is equal to three to the power of three. Lot 17 written as look three 10 equals three 18 close to 11 iss three to the power of seven. What 17 rewrite asked low 3 11 You close to seven a equals a equals 12 IHS equals 23 Power to the 13. What? 17 Be right as law 3 12 equals two. 13 So 80 those 13 equal shoe three to the power of fourth Mott 17. Rewrite us locked 3 13 equals for a equals 14 rewrite. Um sorry a host 14 is equal to three. Power to the nine March 17 rewritten as law. Three 14 E Close, nine 80 Who's 15? Rewrite? Asked. Three to the sixth Ma 17. You write as look. 3 15 equals 26 A closest 16 is able to through to the power of eight watt 17 river. It s law 3 16 equals to eight, and that is our table.

In this question. We're trying to figure out if 450,917 is divisible by three. And to do so, we need to use our day visibility rules which say that if you add all of the digits together, that number that you get means to be divisible by three in order for this original number to be also divisible by three. So here's how that works. We're gonna add together four plus five plus zero plus nine plus one plus seven which if you add them all together you get 26. Now, if you're not sure if 26 is divisible by three, you could just do the process again. To plus six equals eight, which is not divisible by three and 26 is also not divisible by three. Um so that means that our original equation or our original number 430,917 is not divisible by three. And that's our final answer


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