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QuestionMCOsand" ' In Ouacianithen vhalSelecl the correc ansvaer belov:...

Question

QuestionMCOsand" ' In Ouacianithen vhalSelecl the correc ansvaer belov:

Question MCOs and" ' In Ouaciani then vhal Selecl the correc ansvaer belov:



Answers

Solve each of these congruences using the modular inverses found in parts $(\mathrm{b}),(\mathrm{c}),$ and $(\mathrm{d})$ of Exercise $5 .$
a) 19$x \equiv 4(\bmod 141)$
b) 55$x \equiv 34(\bmod 89)$
c) 89$x \equiv 2(\bmod 232)$

So this question we want to solve various mods, mods, concurrences. So 34 x is equivalent to 77 mud 89. Well, so first we'll assume that you know how to find these investors. So in this case, 34 invest is equivalent to 55 more. 89. So this you already found in question six for exercise six. Now multiply that invest on both sides of the equation. You have 34 in verse. That's life very for X is equivalent to 34 34. Invest has 77 mold 89. We know that 34 investors Secrecy 55 investors cancels X is equivalent to 55 type of 77 mud a night. Now, I'll just assume, you know, I have sold this. Um, this number, Yeah. I just assumed that have solved This is 52 more a month. That would be 144 X is equivalent to four. Well, 2020 33. So for this question, the inverse off 140 or invest it's equivalent to, um my 89 89 No. 233 So you could time to investors on both sides. Do you see investors cancels. So all these enormous cancels X is equivalent to 89 times well or more 233 which is just equivalent to equipments 223 mud. She 33 for our last one. C 200 X is equivalent to the team. Look, 1001 So my question six, you have tooth out. Joint invests is equivalent to ni ng 96 more. One on one. Yeah. Um, yeah. Then you have X is just equal to 996 That's why 13 more 11 So this is equivalent to 936 Love 1000 and one.

So since, um, we could, given that x equivalent to one want to And so therefore we can have X can be equals two to t plus what? Where t is just on entered your Since we also have this access equivalent to two months three by substitution, we would have, um, to t plus one is equivalent to too much be sorry for squeezing in there. So now we're trying to Seoul for teeth. So, um so for tea through this equation, we would have t a equals two to mont. I see. So t now equals to, um some three s plus two for esa's, some inter jer. So And we also have X equals to t plus one. So now, by substitution, we would have x then now be equals 26 s plus five by substitution. Uh, so now moving on to the next X is equal. Vaillant Tuesday, 3 March five. Less so for eggs. So be half initially that six s plus five is equivalent to 3 March five This'll eat. Solving for s s would then be equals 23 my five itself. So knowing as we can now form an equation with a new injured. Your K so s would now be equal to five k some against some inter jerk A five K plus three sins X s equals +26 s plus five again. By substitution, we would finally have XB equals to 30 k plus 20. Finally, Now we saw for X is equivalent to 4 March 11. So less soul for K. So again, x iss 30 k plus 23 equivalent to for my 11 We're solving for K. Okay. And what final look finalized Value for K would be 10 ma 11. Okay, so ok can now be again rewritten as 11 j plus 10 Jay again is some intern your so editor. Since we know that X is equals to 30 k plus 23 my substitution, we would have X equals 2 300 30 j plus in Britain 20 c where j is some integer and that is our final answer.

Okay, So if what? This question we want to solve this settle convinces, easing back substitution. So first, let's write this out. Fleets of this is X is equal to do plus by three a Well ace, um, interject Substitute this into this equation. So we have two plus three. A is equal. What equivalent to one? No, Before now it's attracted to to decide so three a is equipment to myself. One No. Four. And you'll notice that three is just minus one for the negative is equipment to one more for a is just equivalent to wasn't municipal. So one model for you can write this out as being one first four B. Um then we substitute this back into the subversion we have. X is not equal to two plus by three. Number one test for so 23 times languages 34 times three gives you 12. B is just by pass by. So we substitute this back into this equation. So we have my class by 12. He is equivalent to three. No fun. Now subtract five toe be is equivalent to my spy, too. No. Five. We divide what's 12 is equal, equal equivalent to two more fire. This is just a doobie is equivalent to myself too. What? Five. At this point, we recognize that the greatest common divisor between two and five is just one. So you can just divide too to be Is this equal to negative off one more why, you should just equivalent to four more why we should just well bye bye type of c c you something today. Then we want a substitute fisting back into this thing. So we have. But he x is equal to five plus 5 12 tons or flight see is equal to fire. Plus totaled four is 48 12 times five gives you 60 what you see by plus 48 easier 53 plus my 60 c So x is just a few moments to 50 really month 60.

Given some information, we have a line, and we know that line is parallel to another line. I mean, this was called I R and the swims MV. And then we have the transfer sel going through there and it transfer. So build through the other way. Yeah. They also tell us that these guys are growing. Now the question is, can you make a statement about two triangles? Um, can we say that triangle? Why are in this king? Grew to another triangle. Which triangle? And how do we know? Well, we know that these two angles are congruent by vertical angle zero. And then we know that these two angles are congruent alternate interior angles here. Um, so then if we line them up, we have angle, angle side as our reason. And you could have gone to this angle over here to doesn't matter with that same reason. If we line them up, starting with I way see that I corresponds with v our corresponds with em and corresponds with him.


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