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Suppose that a and b are integers, = 11(mod19), and b = 3(mod19). Find the integer € with 0 < c < 18 such that_a) c = 13a (mod19)b) c = 8b(mod19)c) c =b...

Question

Suppose that a and b are integers, = 11(mod19), and b = 3(mod19). Find the integer € with 0 < c < 18 such that_a) c = 13a (mod19)b) c = 8b(mod19)c) c =b(mod19) d) c = 7a 3b(mod19) e)c = 2a2 + 3b2(mod19) f)c =a3 + 4b3(mod19)

Suppose that a and b are integers, = 11(mod19), and b = 3(mod19). Find the integer € with 0 < c < 18 such that_ a) c = 13a (mod19) b) c = 8b(mod19) c) c = b(mod19) d) c = 7a 3b(mod19) e)c = 2a2 + 3b2(mod19) f)c =a3 + 4b3(mod19)



Answers

Suppose that $a$ and $b$ are integers, $a \equiv 11(\bmod 19),$ and $b \equiv 3(\bmod 19) .$ Find the integer $c$ with $0 \leq c \leq 18$ such that
a) $c \equiv 13 a(\bmod 19)$
b) $c \equiv 8 b(\bmod 19)$
c) $c \equiv a-b(\bmod 19)$
d) $c \equiv 7 a^{2}+3 b(\bmod 19)$
e) $c \equiv 2 a^{2}+3 b^{2}(\bmod 19)$
f) $c \equiv a^{3}+4 b^{3}(\bmod 19)$

Okay, so in this question, we want to show that if a is, it could be more em in a C is bay Seymour M West is an insurgent. So first off, it's a easy could be more of them than this implies that a is equal to km. Thus be where Kay is some into John. And if we multiply this equation by sea, we have a c is equal to K c M plus my B c So b c just by K c m. Well, we never since C is an inter job and Kay is an inter job with the multiplication off. Two digit is also another interview so we can get this BBC plus by L M where l is even to Casey, which is some indigent. So therefore you have a C is eating too Casey mud and just by definition

Okay, so this question we first have a is for more touching, and B is nine month dirty and we want to figure out numbers, See such a cuz to nine. A more 13. So a easy could perform what? 13 and the mods are equal. So you just start to a as four. So we just have nine times before mother 13 to 36 1 13 to simplify this. What we do is obvious my trip. So the trick that I use is I put 36 over 13 into the calculator and convert this into mixed fashions. So it converted this into mixed fractions. We have 10 over 30 on the 10 Seasons. 10. See Easy to 11 b more 30. So be easy tonight. So 11 times my nine is lining nine more 13. During the central heat, we have 99 over 13. You're betting that into mixed fractions is 78 over 38 is in your model. So you see, C is equal to a plus. But be mug. But see? Hey, easy to four on B 294 plus mine gives you 13 more 13 which is just there, and d we have C are you on the page? See? Is even to choose a place by three b more 13 So we have booked. This is this is hot. So we have a 6 to 4. So this is two times for a possible be easy to nine. So three times line 27 1 13 which is equal to 35. Mom 13 and running 35 with 13 into mixed fractions. We have This is two and 9 13 So this is mine. He sees into a sweat off a beast with mud. That's you. Hey, easy to 44 Squad is 16 plus, but be easy to +99 square. Is it to 81 more? 13. Adding these together we have 97. What's the scene? And 97 over 13 by that into extractions. They should give you seven and six thirteen's. So this is just equal to six now. F c. Is it to ache? You must be huge. More 13 we have. It was just for a few miles on nine Q. More 13. And if you write four Cuban Wiseman, nine cubed over 13 and you convert that into mixed fractions, which is negative 15 and 2 13 But in this question, we have to be a little bit more careful because there's a native. So if we fool you write out this mixed fashions. This is negative 15 minus 5 to 13 so that 15 heat is a question and shoot or minus two will be. Remain at minus. Susan, Remember? So this whole thing, then is even too negative, too more 13 warrior. But if you read the question, we want to say to be listening to zero, listening to say less than or equal to what and because of this, because it's negative to wear outside of range. So to make it come back into this strange, we just add 13 onto this. This is 13 months by two more 13 which is just

Okay, So in this question, you want to show that if a visit to the mod Dempsey is equity D model Emme, then a minus E. Is he going to be my honesty mod m? So, first off, if ace is going to be more M, this implies that a is equal to okay and a plus B. Okay, judges, then from this argument you have see is eager to it's a l m plus de for L in your images, then subtracting this equation and this equation together you have a mon isi is equal to okay and thus be modest, fi l know and saw spiting. Then we take the B and defense. So be be modest by D plus by okay and modest by Bill. And so we're very close now, So B minus g just by okay, months by ill. Yeah, well, we know that since K is an inter job on l wasn't into John and subtracting two insurgents get you another interview. So then this is simply equal to B minus my d plus fight. It say Okay, so let's say he I guess he and where he is equal to K minus five l is in your interviews. So what we then have you have a miner's wife? C is equal to be modesty plus P. M. We're peas. Interject So then, by definition, off your mother, you have a minor. C is equal to B minus by D month and I was wrong.

Okay, so we want to show that it's a divide C B divide seat, and this implies that a B divide CD. So if a divide see this implies that exists, there exists an inter job X, such that a X is equal to see if be divides. The this implies there exists a why being in India, just such that's be why is equal to d'oh. So now if you multiply these two equations together, we have a x be why he's equal to see the and we have a X B. Why is equal to CD reshuffling these arounds? We have a B times by X. Why these people to safety. So now if we let it sit, be equal to X. What's x times? Why? Why? Well, we know that since X and Y are inter jizz, then it also implies that said isn't interject So therefore there exists an inter job zed, which is if two x y, which is in your images such that's a B. Times like that is equal to see d. So therefore, a B divides CD


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