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Problem 4:(a) Find the smallest positive satisfying 14319 = x (mod 4) _ (b) Find the smallest positive such that 3199 = 2 (mod 10) _...

Question

Problem 4:(a) Find the smallest positive satisfying 14319 = x (mod 4) _ (b) Find the smallest positive such that 3199 = 2 (mod 10) _

Problem 4: (a) Find the smallest positive satisfying 14319 = x (mod 4) _ (b) Find the smallest positive such that 3199 = 2 (mod 10) _



Answers

Let $m$ be a positive integer. Show that $a \equiv b(\bmod m)$ if $a$ mod $m=b$ mod $m .$

Okay, so in question, we want to show that if a is equal to be mobbed em, then a more M is equal to be. Well, if a is equal to be more M, that means that exists at Inter Jacquet such that a is equal to okay and plus might be This is by definition, no more now moving be to one side, equal to minus by K. M. Well, now we haven't been to jail here, So you feel that this being a just by Let's say well and where l is its martyrs k, he's an insurgent. So this is what's again, by definition of modern, we have b is equal to a Madi. Yeah, as required.

Problem. The given expression Can be written as one glass 1000 into £10 -4. Yes. 1000 and two 999. You had a type two Into £10 -8? Yes, Just one more time. Have become 1000 C three, 10.10 and so on. Now, this is less than one less. One by 10 plus one by 100 less one by 1000 less sworn Which is equal to one upon 1 -1 x 10. This is a GP. So foster they would buy 1- the common ratio. This comes as 10 by nine, so the interior just greater than the human expression has to be two. So the correct option is B. That's all.

Given the function p of x equals two. X cubed minus x squared plus four X minus two. We want to find the upper and lower bound. Let's go for the upper bound. Starting with one with coefficients to negative 14 negative two. We're looking for closure. Well that is nine negative. Drop down the to multiply add multiply add multiply add all non negative one is an opera bound. Let's go for a lower bound negative one. We're looking for a quotient road with alternating signs. Drop down the to multiply add multiply add multiply add -1. Is a lower bound. Yeah. Yeah. Mhm. So Mhm. Um Let's see if we can find a zero between zero and one By evaluating this function at zero. Um we know that because 01 for X We're gonna get negative too. Let's enter one into the functions would be two times 1 cute minus one squared Plus four times 1 -2 which is positive three. So we go from a negative to a positive between zero and one. So there must be zero between the in the interval 0 to 1. So using the bio section method will make a little table. Yeah. Yeah. Mhm. Yeah. We already know that from 0 to 1 that PJ was negative. NPV was positive. So let's find the nearest halfway between zero and 1 0.5. Let's find the sign of 0.5. Yeah. Yeah. Mhm. It is zero so there is no sign. We found the zero. It is at 0.5. Yeah.

Given the function p of X equals X. To the fourth minus two. Expert minus three. Want to find the upper and lower bound. Let's find an upper bound. Starting with one with coefficients 10 negative 20 negative three. We're looking for a quotient row that is all positive. To drop down the one multiply add multiply add. Okay so it's not 1 42. Drop down the one multiply add multiply add multiply add multiply add two is an upper balance school. For a lower bound. We're looking for an alternating sign. In our culture mode. Drop down the one multiply add multiply add two negatives in a row. Let's go to -2. Yeah. Drop down the one multiply add multiply add multiply add multiply add alternating signs negative two is an up lower bound. So our upper bound is positive two. Or lower bound is negative two. Let's try to find a zero between one and two of two is an upper bound. So we can go to a calculator and input one. So if you want to the forest -2 times one Squared -3 that's negative four ps two is positive five. So because there's a zero between negative four and five and their continuous there's a zero between one and two in this interval. So now we can go to the by section method. Really little table. Mhm. With the interval 12 we already know that P. S. A. Was negative and PBS positive pathway between 12 is 1.5. So let's see How does this function evaluate at 1.5? It is negative which means there must be a zero between the midpoint and be. So what between 1.5 and two which is 1.75 So we'll evaluate at 1.75 and we get a positive output which means the zero must be between Um 1.5 and 1.75. Halfway between 1.5 and 1.75 Is 1.625. Okay. Which is a negative output? Which tells us that zero must be between 1.625. At 1.75, Halfway between the two is 1.6875 evaluate that's a negative output. Which means the zero is between 1.6875 In 1.75 To the nearest tent. That's 1.7. 1 point is we have to keep going Halfway between the two mm at one point 71875 mm. Mhm. Which is a negative output. So it's now between 1.71875 and 1.75 Halfway between the two. There's one point 734375. Okay. 734375, Wikipedia. Yeah. Which is a positive output. Which means we must have a zero between 1.71875 and 1.734375, which we can stop now because we're to the nearest 1.7. Yeah.


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