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Use Euler's generalization of Fermat's theorem to find the remainder of 7 when divided by 24. In Exercises 14 through 16, describe all solutions of the gi...

Question

Use Euler's generalization of Fermat's theorem to find the remainder of 7 when divided by 24. In Exercises 14 through 16, describe all solutions of the given congr Examples14. 45x ≡15(mod 24)16.41x ≡125(mod 9)

Use Euler's generalization of Fermat's theorem to find the remainder of 7 when divided by 24. In Exercises 14 through 16, describe all solutions of the given congr Examples 14. 45x ≡15(mod 24) 16.41x ≡125(mod 9)



Answers

Use Fermat's little theorem to find $7^{12}$ mod $13 .$

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?

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

Okay, so this question is asking us to use the remainder theorem to find the remainder of s so forth minus nine x squared plus 14 divided by X minus two. What the remainder theorem says, is that if you have a polynomial f of X that if you divide it by another binomial X minus K, where Kay is a number, then the remainder is Ah, leader is f Okay, whatever value you get when you plug in K into your polynomial And so our problem well, in this case is X to the fourth minus nine X square, Uh, plus 14. Uh, And if you plug in K A r k. And this problem is so X minus K and R A k r binomial that we're dividing by X minus two. And to get this to look like that, then I k must be positive, too. So you plug in a positive to into our Paula No, mule. We get to to the fourth minus nine times two squared plus 14. And when you reduce this, this becomes negative. Six. So that's your remains.

This problem. It wants us to use the remainder theorem to find the remainder. So we know they're made of Euros says that if we have a polynomial were divided by X minus K, then the remainder is just That's okay. So all we have to do is we see we have a exclaims, K form years. That's why it's too. So we know K equals two. We just plug that into F. This is F We're gonna get our manners. F of two is to to the fourth minus nine times to the square to square, which is four plus fourteen's. We have 16 minus 36 plus 14 in this equals negative six, so our remainder is negative six.


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