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B1. a. Construct the Addition Table of Congruence Class𝑍5. B1. b. Construct the Multiplication Table of CongruenceClass 𝑍5. B1. c. ...

Question

B1. a. Construct the Addition Table of Congruence Class𝑍5. B1. b. Construct the Multiplication Table of CongruenceClass 𝑍5. B1. c. Using Euclidean Algorithm, find the Inverse of theinteger 13 mod 19.

B1. a. Construct the Addition Table of Congruence Class 𝑍5. B1. b. Construct the Multiplication Table of Congruence Class 𝑍5. B1. c. Using Euclidean Algorithm, find the Inverse of the integer 13 mod 19.



Answers

Find an inverse of $a$ modulo $m$ for each of these pairs of relatively prime integers using the method followed in Example $2 .$
a) $a=4, m=9$
b) $a=19, m=141$
c) $a=55, m=89$
d) $a=89, m=232$

Kowtow to fight the equivalent classes of all inside Congress caused the inside the congruence e module. Five. So we're giving, we're told, defying the equivalent clause of to well to the inside of a couple of classic two. That means your remainder one divided by five has to be the same as choose remainder. So a an element a is inside the current class off to if a mod five, If you two more five Q Or, in other words, a in the Congress cost in a congruent one. In module five, we can get the same thing for three. So the is Congress to three. March 5 implies that AIDS in the equivalent classes dream and likewise for six A is in the cruelest cloth 68 Congress to 1 March five left because six months he was one. And here we have my street, my five a minus bi or two. So a is inside the equivalent because of my ministry. If a Congress to my street or to mark five

Everyone. So this problem is fairly similar to problem five. So I'm really gonna go through the answer, so a equals two, um, M equals 17. This is for part A. We need to find we need to first perform the Euclidean algorithm. Um, which is 17 equals to eight times two plus one. Don't forget the remainder. I have been there. Two equals 22 times. One. So the greatest common division is, um, it's one of A and m iss one. So now we need to find greatest common division as a multiple off A in them. So we see that greatest common division is equals to 11 can be C b. You simply reading written as 17 minus eight times two or one time, 17 minus eight times you. So are a is, too. So the inverse number ISS minus eight. Um, we have minus eight. My 17 similar to nine months. 17. So therefore, nine is also an inverse of to move on to be so for B, we have a So are a equals two. 34 m equals 89. Um, so of course we perform the Euclid. Dine out for them that 89 equals to two times 34 plus 21 34 equals to 21 plus 13. 21 equals two 13 plus eight. 13 equals to eight plus five and equals two. Five. Plus the five equals 23 um plus two. Three equals to two plus one, and one would be our greatest common division. So again we find the greatest common division as a multiple Off A and M rated comment division. ISS, um is 34 off 34. 89 equals to one. We need to write it as a number A number times 34 plus a number times 89 that equals one. This is what you need to right down. So the answer for this one would be 13 times 89 minus 34 times 34 sis equals one. Um, so we need to find an inverse of a A is 34. So the inverse of a would be minus 34 so minus 34 my 89 this is similar to or is equal to, um 55. My 89. So 55 would be our inverse of coefficient. So for part C um, we have a equals one for four m in close to 233 and find their greatest first task. You need to find their greatest common division. Um 2 33 is equals to one for four plus 89 144 equals 2 89 plus 55 89 equals two. 50 who 55 plus 34 55 equals to 34 plus 21. 34 equals to 21 plus 13. 21 equals 2 13 plus eight. 13 equals two, um eight plus five A is equals to, oh, a equal. It would be equal to five plus three and five would be equals two three plus two, three is imposed. True two plus one. Yep. So their greatest common division IHS again one. So now if I need to find ah multiplication of, um A and M so that the result would be equal to one. If I were to written down like a general rule, it would look like a a number times a plus a number times be a number time. Sorry, m 2 33 equals to one. So in this case, this valley here would be eat close to 89 in this value here would be equals two minus 55. So therefore, the inverse of coefficient A in this case is baby fine. For part D, Um, A is equal to 200 m is equals to 1001. Again, we apply or perform the Euclidean algorithm in which, um, we have 1000 and one would be equals 25 times, 200 plus one. Ah, so lucky. So we don't have to do that long lady stuff. So 200 again equals 200 times one. So the greatest come the division in this case is again equals one. So we need to find him. Pull off A and m such that is equal to one. So a number times 200 plus a number times 101 IHS equals to one. Um, so you guys, this value would be equals to one. And this would be equals two minus life. Because eso we have minus five My 1000 and one iss similar to 196 MyDD Did I just say 100? 101 equals 2 90 996 months. 1000 and one. So the inverse of a ISS none written. 96


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