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Use Fermat' $ theorem t0 calculate 574536T8 mod 11Use Fermat 5 theorem t0 calculate 4 "modllCalculatemod 12...

Question

Use Fermat' $ theorem t0 calculate 574536T8 mod 11Use Fermat 5 theorem t0 calculate 4 "modllCalculatemod 12

Use Fermat' $ theorem t0 calculate 574536T8 mod 11 Use Fermat 5 theorem t0 calculate 4 "modll Calculate mod 12



Answers

Use Fermat's little theorem to find $23^{1002}$ mod 41

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.

Okay, so this fishing want to keep you Paris Months surfaced in happy is far to the 2000 and three more seven. So what we do know what? So you want to write this in terms of 6 10% number, so have 2000 and three is equal to 334 times six months. So we can write. This number is being fired. Six All this to the 334 multiplied by five minus one. Mom. So bye. Fermat's last year. You know that this number is one. So 12 334 students and one to this number. You just left this five in verse. What? Seven. You know, we know that five investors just equal to three months. Seven. So our next one is Bob's, the 2003 more 11 I wanna live in. So once again, we want to do the same thing, but in terms of 10th. So we know that 2000 and breezy to 200 times by 10 plus three. So you can write This number is being far too 10 that she 100 times five here more 11. This now becomes one So this is five. Q five is just one of five more 11 side. This is just for 11. Well, you know, and our last 112 2003 More 13. But the same thing with 12. So we have 2000 really easy with you. 167 times by whoa. Modest one. So then we can write this number. She as being boxed the 2000. Think about it. What's 12 to 1 67 as a five minus one more 30. A famous last serum. This becomes one sense is why invest Mole 13. And this is just eight more busy now for the next pot. One of these charms remain in theory, into flying by to the 2003 more 1000 and one. So we know that one doesn't have one is equal to seven times by 11 times. So I really need to do Is this soul on what we have from here? Pay this question that fight to the 2000 and three. If I just let this be eggs, we know that X more. Most seven gives you three x more 11 days. You four and x modern more 13 gives you a So this is China's remaining serum to solve this? No. So how did this? So we really to find seven times, like 11 times, like 13. So this is 1001. Um, they need 1000 and one divided by seven. So this is just 143 3 1001 divided by 11 is just money one and 100. 1001 Divided by 13 with the 77. Now, we need to find 143. Inverse, most seven. It just is a calculated for this. Just five. My one ingress more. 11. This is just four. And I'm 77. Inverse mold. 13. This just thinking, Paul. So now about Hein so much. I never made it there. The answer. Just be three times 100 43 5 plus four has anyone for plus 877 So we write that three times. 143.5 plus four times by 91 tenant only one time full plus eight times for 77 times talk. I'm just gives you 10,000. Monitor. Miami Three more One. There is one which is just equipment to know. 900 and a three. Mole 1000 and one

We would call. Okay, One equals one in two equals two. A three equals three, a four equals four and and one he calls too. N two equals three and three equals five and four equals 11. And now we have and IHS a month a total multiplication of and one all the way to and four in this case. So we would have two times three times, five times 11. All of this equals toothy and written 30. Now, our next step is to find Z value. So see one as equal to and over and one C two equals two and over and to C three iss on over and three se four equals and over end for so n is thean written 30 over and one is to this equals to 165 Same things He too would then be wondering in 10 c three s 66 c four s 30. So now we have ours rz or a r n Next we need to find our y value. So why, um why one b equals to my see one? Uh my and one in this case would be the one is 1 65 months and one month at one is to what? To this will equals to one. Same thing. Why, too, would be one Dritan 10. What three equals to two? Why three equals to 66. My five months. Five. It was 21 Why? Four equals two, um, 30 rate 30. My, it's 11 equals seven. Yep. So our next up would find W Valium, so w would be equals. W one equals two. Why One time z One is why once he won iss. Why? Once he won Mott. Sorry, but, um does in close you wonder it in 65. Uh, my, the in Britain 30 equals to wonder. In 65 same method W two would then have a value of 220 w three within half value of 66 w for would have value of 210. Your race so that we have some space to find Capital X. We would obey this rule. So X, with equal with is equivalent to a one w one plus a two w two plus dot, dot, dot to a four w four. All of that. What? Take a module off and so if we pluck the number in, we would have 165 plus two time 220 plus three times 66 plus four times 210 all of that modular for soon written 30. So therefore the X would it would be equivalent to 1643 my but they didn't written 30 and this has a value of the remainder of this would be 323. So therefore all solution should be taking in the form general solution it is taking in the form 3 23 plus 30 33. 30. Multiply it by some random K, in which case a is an arbitrary cherry interview.

Okay, So this question we want to compute various on various numbers. I guess so. Three Rio too more by So to compute this, we use so much little syrup. So by famous little serum three to the four is equivalent to one month flight. So we have 3302 is equivalent to 3 to 4 times less 70 by plus fire to this is quote money. Why, This is just going to be, um, reserve board. Traversoni by has a three squared more five. And this becomes one said this is just three swift. So nine more five, which is just four months by now. 3302 What's seven? We did the same thing by Fermat's little serum. We have three sixes gripping to one most seven now three to the 302 is equivalent to three to the 50 times last six suss fight too. So then this is almost seven. So then this is equivalent to three to the 50 tons of six times by three square marks. Seven becomes one bite for a single theory. So this is just nice. Boys in line, Mom seven. This is just too months. So now for the last one, we have three to the 30 shoot. Uh, what? Seven. So, by Thurman, it's little theory. We have 3 to 10 is equivalent to one. Well, you let me so rewarding this. We can rewrite this as, uh, three to the three times by 10 suspects, 100. But say times 10 that's what to it's equipment, too. Read, uh, 10 time by 30. That's what you tied by three squared Mark Levin. This becomes one. So this is this mine say is equivalent to nine monkey. Now, what will use, though, is for 300 truth in what five will use that this is equal to Meyer for 302 months. Seven. We use that this is equal to nine. And for 302 more 11. We used it to keep it in mind. So in summary, then what we have is 3302 is equivalent to nine. Well, why three to the 302 is equivalent to nine. Well, seven and three to the 302 is equivalent to nine model. So by the Chinese remainder theorem, what we have is three to the 302 is equivalent to nine more by seven times, so this is just fine more 300 and 85.


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