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2. (4 points) Calculate gcd( f (r), g(-)) for f(c) = ~' Ql]2 and g(r) = r +2 +I+1...

Question

2. (4 points) Calculate gcd( f (r), g(-)) for f(c) = ~' Ql]2 and g(r) = r +2 +I+1

2. (4 points) Calculate gcd( f (r), g(-)) for f(c) = ~' Ql] 2 and g(r) = r +2 +I+1



Answers

Use the Euclidean algorithm to find
$$\begin{array}{ll}{\text { a) } \operatorname{gcd}(1,5) .} & {\text { b) } \operatorname{gcd}(100,101)} \\ {\text { c) } \operatorname{gcd}(123,277) .} & {\text { d) } \operatorname{gcd}(1529,14039)} \\ {\text { e) } \operatorname{gcd}(1529,14038)} & {\text { f) } \operatorname{gcd}(11111,111111)}\end{array}$$

All right. So here, using the you could in algorithm to find greatest common advisors and the greatest common divisor, the pretty Adam states that we're finding the good is common divisor is gonna be the same as taking the greatest common divisor of these smaller of the two numbers and the remainder. So that's used their examples to sort of walk through it that really looks like and so initially were given to find the greatest common divisor of 12 and 18. So this should be equal to taking the greatest common divisor of the smaller one, which is 12 with the remainder rights. When you divide 18 by 12 you get remainder of six. Okay, now I'm thinking the same thing. So this should be the same as taking the greatest common divisor of six. Then we devised expect 12 remainder zero. Well, so now now we just get this is equal to six, so it can't go down any further. So let's try B here. We want the greatest common divisor of 111 and 201 so this is gonna be equal to taking the greatest con divisor. We'll take the smaller 1 111 and the remainder when we divide this. Right. So that would be 90? No, do the same thing. This will be equal to the greatest common divisor of the smaller 1 90 and the remainder which is 21. Okay, which is going to be equal to taking the greatest con advisor? We'll start with a smaller 1 21 and the remainder when we divide it into 90. So 90 divided by 21 gives us a remainder of six. The same thing again is common. Divisor will take the smaller 16 divided into 21. Because this remainder of three Okay, this is equal to taking the greatest common divisor. We'll take the smaller 13 When we divide six by three, we get zero. Okay, so that gives us a three for our solution of the greatest common divisor. I guess we'll go to a new page here just in case this one is long on for part C, we want to find the greatest common divisor of 1000 and one and 1 13 31 Okay, so we start by taking thes smaller number, which in this case, is 1001 you would divide 1331 by 1001 and we'll keep the remainder in this case is 330. So now we'll take the is common divisor of 330. We'll divide 1000 won by 330. Thank you. Three main gear, which is you live in. So now we'll take the day's card advisor this morning. One. Divide it 330 By 11 we would just be 30 and zero. So there was no remainder here. In this case, Our greatest common divisor then is just 11. Okay, so to go to part D, we want the greatest common divisor of 12345 and 54321 Okay, so it's gonna be equal to taking the greatest common divisor of these smaller one 23 45 Then we divide 54321 by 12345 and get the remainder, which is 4941. And so that is equal to taking the G c D of 4941. Divide 12345 by it. Then the remainder is 2463 In case we follow, the same thing will take the smaller 1 2463 Divide 4941 by it and get a remainder of 15 Nice and small. OK, so we take the greys count adviser of 15 divide 2463 by it. What's left is a three. Okay, so then we'll take the greatest common divisor of three. Divide 15 by three with the remainder zero. So beckons us a three. Is our gaze common divisor? Not particularly great. But there is nonetheless so well over half way there now. We want the greatest common divisor of 1000 and 5000 and 40. OK, so we take the smaller one. It is common divisor of 1000 uh, divided by 5040 And keep the remainder. It's gonna give us 40 so this is gonna be equal to the girl's common defies. Er 42 fired 1000 by 40 and we actually get a remainder of zero. Okay, so that means that 40 is their gaze common divisor influence. We just have one left here for F here. You want the G C D of 9888 and 6060. Okay, so you know, that's equal to G C D of 6060 plus the remainder we divide it, which is 38 28. Okay, so it's kind of equal to the G C D of 38 28 and 20 to 32. So then we take the greatest con adviser of 2232 the remainder when we defied 3828 by it, which is 1596. Okay, so that should be equal to the Greys common divisor of 1596 and 636 which is equal to the greatest common divisor of 636 and 324. Okay, so it's gonna be equal to greatest common divisor of 3 24 and 3 12 which is equal to the greatest common divisor of 3 12 and 12. Okay, which is equal to the greatest coming adviser of 12. Yeah, and zero, which is just equal to 12. All right, so that concludes the process for using the Euclidean algorithm


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