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Solve thet linenr congruency (mod 19) Note: Prouide all details of vour computateon} Frovide your' anSuxr 4s Tluttibet between zero and 19...

Question

Solve thet linenr congruency (mod 19) Note: Prouide all details of vour computateon} Frovide your' anSuxr 4s Tluttibet between zero and 19

Solve thet linenr congruency (mod 19) Note: Prouide all details of vour computateon} Frovide your' anSuxr 4s Tluttibet between zero and 19



Answers

Find the solutions of the congruence $15 x^{2}+19 x \equiv 5$ $(\bmod 11) .[\text { Hint: Show the congruence is equivalent to }$ the congruence $15 x^{2}+19 x+6 \equiv 0(\bmod 11) .$ Factor the left-hand side of the congruence; show that a solution of the quadratic congruence is a solution of one of the two different linear congruences. $]$

Okay, So this question we want to find what is four ex government by mod night. So we want to solve for X using in verses. So the first thing that we know is that the inverse off for mother mine is eating too minus two mud nine music seven. What Night. So if we times the inverse of both sides of four in first handler for X is equivalent to four Inglis time bye mode? No, the investors goes away. Say left with X is equal to We can take our pick of the litter here so you can either choose minus three or seven. I think minus true will be the easiest. So it will be minus two times but five. So that's negative. 10 more mine. Then this will be equal to minus 10 plus by nine month nine so that this is equivalent to minus one Mud. Nine m. King added nine months against minus nine months. No, I So this is just equal to eight. Good night.

So, um, since we know that 35 is in closed to five times 10 7 um so therefore, this means that square is a violent too 29 the prevailing too for well five. So therefore X is equivalent to two what five or ex prevent to minus 2 March five. So that's the first case for five second case for seven excuse equivalent to 29 equivalent to one mud. Seven. This mean that X can be equivalent to minus one. What seven or X can be a privilege to one? Yeah, one month, seven minus one with seven in one month. Seven. So now combined, the solutions by the Chinese remain earthy room. Recall that if X is equivalent to a mod five, then X be equal to five K plus A for some inter jury K. So if X is a equivalent to Aamodt seven, there's also become five K plus a local village to be but seven. This turn. This will turn into tu minus two K that prevail it too. B minus a mont seven or two. Okay, equivalent to a minus beam on seven, since they can be either two or negative too, and B can either be one or negative one we have ourselves in. We have our self total Four more case to consider. So, case one A, um a is a village to be is equivalent to one. So we now have two K is equivalent to ask you recall a minus bi mont seven, which is equivalent to minus three. What we know Tu minus one is just one my bet. One, not seven IHS, he goes to for months seven and we can rewrite that Us. Okay, with them equals to seven n plus four for some entered your end. So we know previously as well that X equals 25 K plus a and therefore X This equals two 35 plus 22 in which a is EPA Vaillant or equals 22 So that out that's our first case. Our second case IHS a equals two b equals minus one. So tu que equivalent to a minus b a minus bi iss three Mont seven. This mean that que is equivalent to five a lot seven. Then we would have k equals to seven and plus five for some entered your and and hence we know that X equals five K plus a equals five K plus two. We replace okay in this to this equation. Then our final X answer would then be 35 plus 27. That's our result For second case, our third case, this 18 is equivalent to minus two, and B is equivalent to one. So we have two. K is equivalent to a minus. B in this case would be minus three months. Seven equivalent to four months. Seven So therefore que is equivalent to two months. Seven. We can rewrite K ass seven and que equal seven n plus two for some inter jerk. And therefore we would have X equals two by replacing K with the ex, um, formula abo we would finally have X equals 35 and plus eight for case for we have a equivalent to minus two. Be equivalent to negative one. We have two K equivalent to a minus bi, which iss minus to plus one is minus one. I just realized something in my solution. Um, so case they, as you can see, a A minus B would be minus two minus one would be minus thing there is not supposed to have Aamodt seven at the end, so it would just be minus three, equivalent to four months. Seven. So going back here, case for um to K is equivalent to a minus. B minus to plus one is minus one minus. One is equivalent to six months. Seven. Therefore, que is equivalent to cream lots seven. So K can be now, um, right in the formula seven and plus three for some interject and and hence we know that X equals five K minus two. Then our final answer for X would be 35 plus 13.

Okay, So this question we want to solve the congruence X squared is equivalent to 16 more 100. And so the most obvious attitude this is X is equivalence to possum eyes for more life, which is just full and next 401 more. 105. I think this really pretty obvious. Yet now another answer comes on. The sole exploit is equivalence to 16 more three, which is equivalent to one. I want one more three result. Expert is equivalent. Just sitting most fine, which is the same thing as buying one. Fine. I've got his facts wrong about this without a point. So exploit is equivalent to 16. What? Seven. Which is the equivalent to two 17 Didn't we solve this? Using the Chinese? Remain out there. So we have one. Well, we we have one. Well, fine. And to Well said, It's interesting that we have is three times five times ms seven. He's 100 and five when 105 by three gives you, um, you 35 105 by five is 21 105 to 7. Your people. Now we want to check with Eve us is safely in verse More. Three. This is equivalent to to give us more three, which is just too. They're one of Catherine 21 In this more why was just one and 15 ingress more. Seven. Which is responsible then China's remain Ethereum means that we most apply 1 35 too. So one time they got to who plus one times 1 21 and one was fun. Times 21 1 Plus she has one. Do you talk about 15 times? One it gives you. This is 70 plus by train one. Spy 30 which is 100 points so we can write this then, as explained, is a good ones to 20.1 more. 105. My original ones is 11. So X is also two plus with my fork rivers personalized 11 more 100. So reality we have plus or minus 11 and plus alarms for we'll get a flight. I'm like what the question says, Well, that's it

So you want to use finesse that will threaten to find what this number is right here. So the message with syrup as it relates to this means that 23 to the 41 miles by one So that would be 40 is equivalent to one more one. So this is just a mess. It'll soon So what? What this means is that we want to write 102 in terms of 40 as much as you can. 23 to the 1002 is equal to 23 times by 40 times by 25 plus bite you. So have I got this is if you do floor 102 divided by 40 40 we should get 25 and the two not to come back. So now we can write this as being 23. 23 squared times by 23 to the 40 as a 25. This is 23 squared times 23 40. And then I would write this as a power 25. Now, if we take more 41 we know that by her master off here in this number, this number here becomes one so want to 25 gives you one. So this is equal to 23 squared mother. 41 boy triplets. I should say evil. And this this is equivalent. Well, it Mrs Equal pick over that same thing. Five to mine more. 41. So that's just squaring 23. And if you actually take them out of this, this should be equivalent to 37. Mother, what do you want?


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