5

6x = 15 (mod 21).36x = 8 (mod 102)....

Question

6x = 15 (mod 21).36x = 8 (mod 102).

6x = 15 (mod 21). 36x = 8 (mod 102).



Answers

Find all solutions, if any, to the system of congruences $x \equiv 5(\bmod 6), x \equiv 3(\bmod 10),$ and $x \equiv 8(\bmod 15)$

Okay, so in this question, we want to solve this system off Concours. So pressing that will do is we fully right At this model. The X is equal to three plus six a where a is some interview now what we'll do is we'll have well, substitute this into this. More question. So three plus six A is equivalent before more. Seven. Subtract Street six is equivalent to one. Oh, we'll soon Now that's six months. Seventies mice one. So this is making a is equivalent to one well seven. And so a is equipment to minus one, which is just six months in. So what this means is that a is equal to six plus five seven. Let's say B for the something to job. And if we substitute this into this equation, we have X is an equal to three plus by six and six plus seven Big. This is three plus two countries is 36. Accept 70 42. That's what for you to be. There's express treated you 39 Well, that's like for you to be. And by definition of mud, this is just equal to 39. One for you too.

I get. So we want to solve the concurrence relationship to excess equipment to seven mud 17. So it just all this we used the investor to invest is equivalent to nine more 17 now solving this All we did is we take will not apply to invest on both sides. Did you invest time by two ex is equivalent Thio seven type of two in verse More 17 chain gets his equal tonight with the two of us is castles So exits the equipment Thio £9.7 which is 63 63 model 17 No, we have first we want to find 63. You got about 70 Florida's. So this is it. 23 And we have 61 have 17 plants were three is equal to just have 63 is even to three tons of 17 plus by Flo 00 So heads 63 months 17 This one cattle leaving you tell that this is just equal were equivalent to 12 Move 17

Here three plus X is equal lento. Okay to March six. Right? Which implies three X plus x minus two years. Divisible by six Or 1-plus 6 is divisible by six. Yes, Putting the value of X which is doing in the options 1, 3, 4 and five. Yeah. We get the value one plus access 2, 4, 5 and six. And the options, which is correct among the forties. Mhm. Only the 4th 1. So our value of X can be fight. Yeah.

So to solve this equation, My goal is to get experts. Self X is currently being multiplied by six Soto. Under this, I would divide by six on the left because then my sixes would cancel. X is no isolated, but I have to make sure that I do this on both sides. So because I divided the left by six, I also need to divide the right by six and 18. Divided by six is three and so that would be my solution. And I can check this by doing six times three, which is in fact 18.


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