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3 _ For how many positive integers m; 23 = 7 (mod m) is true? A) 3 B) 4 C) 5 D) 6 E) none of these...

Question

3 _ For how many positive integers m; 23 = 7 (mod m) is true? A) 3 B) 4 C) 5 D) 6 E) none of these

3 _ For how many positive integers m; 23 = 7 (mod m) is true? A) 3 B) 4 C) 5 D) 6 E) none of these



Answers

Decide whether each of these integers is congruent to 3 modulo $7 .$
$$\begin{array}{ll}{\text { a) } 37} & {\text { b) } 66} \\ {\text { c) }-17} & {\text { d) }-67}\end{array}$$

Eso to watch the three x plus one conjecture, also called the Cola's Conjecture. One simply starts with the number at hand. Onder follows the steps quite algorithmic Lee. So let's start off with six sixes even as we divide by two. That gives us three threes. Art eso three times steepest ones 10 and you will continue the steps. So 10 divided by two that's five and then five times 3% is 16 now e stopped 16 because 16 notices. That's That's two out of four on, basically a soon as you've hit a power of two, you know that you will reach one. The reason being that to dip our four is even so that gives you you divide by two. Andi, uh, to to power for divided by two is two to the power. Four minutes one which is two to power three. Right, So another way of saying that's just eight on bond. Eight. Again Is even Andi dividing to depart three by too busy to the path to just four and so on. So forth um etcetera, etcetera on. You know, you can generalize this as soon as you hit to the power n Then you get to to the power and minus one, etcetera, etcetera, until you reach one. And obviously, as soon as you get you hit one, um, you go back to four. And then two and then one, etcetera. Etcetera. Okay, Now, the chain for the second number is slightly, slightly longer, but, I mean, it's the same process, so seven is odd. Um, Times three plus 1. 22. Do I buy two? That's 11th. Um, that's 34 then 17 and 34. Divided by 2. 17, 17 times three plus one gives you 52. Do I tried to. That's 26. And then again, the all party that's 13, um, 13 times three plus 1 40 and then notice. Um, well, first you reach 40 than 20 right? And then 10. And then notice that once you reach 10. Okay, You're basically at this step here has psychic. You're basically at this step here off the first chain. Okay, so you know that once you reach 10, you actually go to five. Okay. And then 2, 16, 8 to 1. Okay. So you can already stop once you get 10. Because you know, from the previous example that you will reach one and therefore 4 to 1, etcetera, etcetera. So that's gives you one. And the next two examples are actually very simple because you've actually already done them If you start with 17. Well, in the previous example, you've reached 17 here. So you know that you will get following this chain, you will obtain s Oh, sorry. Following this chain here, you obtain one at the end of the day. Um, so that makes your life quite easy. And then four. Question D uh, you have 21 to enjoy, which is odd. Times 3 63 plus 1. 64 on board 64 is to the power six. Andi, using generally is quite useful to know the different powers of two. Um, Andi, as I said before to the party and necessarily leads you to one. Okay, hope this out.

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.

The given equation is and minus five plus minus M minus, toe iota equal do three minus seven iota. We need to find the values off and on m such that this equation is true in orderto find these values. We quit the real parts off the equation under imaginary parts of the equation. We get too, and minus five is equal to three on equating the imaginary parts we get minus M minus. Do is equal. Don't minus seven. Now he's sold these two equations to get the values off and on them. So we sold the first equation to get the value of pen. The ad five on both sides of the equation and get to N is equal to eight, dividing both sides by two. We get an easy call to four on. Now we saw the second equation to find the value off em. We multiply both sides of the equation by minus one and get em. Plus two is equal to seven. Now these have tracked two on both sides of the equation and get M is equal to seven minus two, which is equal to five. So the values off him on and that makes the question true are M is equal to five on and is equal to four, and this is the answer.

It's so in this question, we want to show that a is equivalent to be more em if a more M is going to be so. We first start up with this statement and one produces what is what he's saying. So to use the statement well, first was the first part of it. So if we can, we know that we can write a easy to see m suspect be where? Sea and the Vintages. We also know we can write be as being e m plus by f where e and f r vintages. So by definition, off mods this first month is just beat the second What is this, Steph? So day is going to now If we saw four day we have be easy to a miles by C m. We have f busy with you ever be monster he and so using this equality we now have a month of C M is going to be modest. We'll see now it was attract be across so they must be using. This is a e e plus See America the CM must e m so we can talk it out and we just see my sweet So what we know is that C and E are indigenous. So when we subtract him, this is just another interject here. So this is m tongue. When it does our music to negative a Mars will be so since we have m tired by Inter, what this means is that M is the divisor off a man despite be so now since any devised by a B, by definition, off con fluency eighties equivalents to be more and said this implies this.


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