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A. let m be an even natural number. prove that 7|(10^3m -1).b. let a and b be natural number that leave a remainder of 4when divided by 9 so that 9|(a-b)....

Question

A. let m be an even natural number. prove that 7|(10^3m -1).b. let a and b be natural number that leave a remainder of 4when divided by 9 so that 9|(a-b).

a. let m be an even natural number. prove that 7|(10^3m - 1). b. let a and b be natural number that leave a remainder of 4 when divided by 9 so that 9|(a-b).



Answers

Show that if $a$ and $m$ are relatively prime positive integers, then the inverse of $a$ modulo $m$ is unique modulo $m .[\text { Hint: Assume that there are two solutions } b \text { and } c \text { of }$
the congruence $a x \equiv 1(\bmod m) .$ Use Theorem 7 of Section 4.3 to show that $b \equiv c(\bmod m) . ]$

So we're in heaven M s a positive integer and six plus one Gulf and plus one 18 plus one are old crime. We need a proof and you close to six m plus one time is 12 plus one times 18 plus one by distribution we with by distributions and multiplication of these old brackets. Together, we would then have 1296 m que plus the 96 EMS Square plus 36 m plus one. So we now apply exercise there. Um, exercise 48 that if six. M 12 m and 18 m r division er's off n minus one so and minus one then equals two 1296 m u plus the +96 m Square plus 36. So common factor we would have who common factor that we have 18 m time. 72 mm square plus 22 m plus two a divide be if they are introduced C that b equals A and Z. So from this we can see that six, um divides a n minus one. 12 AM also divided by an who and minus one and 18 and can also be divided by and minus one, we can now apply the result of exercise. 48 apply exercise 48 result. We then have that and ISS a Carmichael number.

So in this problem we have been given that use nuclear division and not to show that the cuba of any positive integer is of the form 1919 plus one or nine plus eight. So let's start let yeah me any positive indigent. Let a be any positive in danger and be Is equal to three. Now using Euclidean algorithm is equal to that implies A is equal to treat you plus are really Q is greater than equal to zero and R is equal to uh yeah is equal to zero one and 2. Because artists since art is between zero and is less than three. So now therefore is equal to treat you or three Q plus one or treat you place too. Know therefore every number can be represented as these three forms. Now there are three cases. So case one would be so case one. So in case one A Q. Is equal to Treat you to the power of three. That is equal to 27 cube cube. That is equal to nine and 2 three cube cube. That is equal to nine. And all right. Where it means? And Indonesia the immigrant and teacher. So today a musical too three cube cube. Similarly in the 2nd case. So case too. So here is equal to trick you. Place one. So therefore a cube is equal to Take you place one whole cube which is equal to 27 QQ Plus 27 to square plus nine Q plus one. So therefore it would be equal to nine and two. Three cuba cuba. Best trick you squared plus Q plus one. So therefore it would be cool to nine n plus one. All right. Thank you. Yeah. M. S. So where it means? And in teacher such that So very amusing. It is our sister to music. Were too treat you cube. Let's treat you square plus you. All right Now, in the Turkeys there this case three. Therefore, in case three we're is equal to 3 2 Plus two. So, Cuban both sides A cube is equal to three Q plus two whole cube. All right. So, therefore it would be equal to 27 Qq. That's 54 q sq. Just 36 Q plus it. All right. So therefore this would be equal to nine and 23 QQ. There are six. You're square. That's for you. And then out of the bracket plus eight. So, therefore a cube is required nine plus eight. All right. Read M. S. And in danger. So, today. Amazing. So where amazon in diesel? Such that such that the music will do three Q cube please? Excuse square. Let's forget you. So, therefore, it means therefore the cuba of any positive indigenous of the form nine AM nine mps, one or nine plus eight. So, this is the answer. Thank you french. I hope you like the video.

This problem we have been given that you can use nuclear division member to show that the square of any positive integer is either a form three um or three investment or for some in teacher. And hint has been given that let X be a any positive in teacher then it is of the form 3232 plus one or 32 plus two. Now square each of these. And show that they can be re written in the form of three or more three and plus one. So now let A. B any positive integer. This transfer positive. So let A be any positive integer And B is equal to three. So therefore by nuclear Lamar is equal to and therefore he is equal to using the nuclear algorithm is equal to thank you. Place are for some interior Q greater than equal to zero and R. is equal to zero 12 Because our lives between zero and his resident three. So now therefore is equal to 32 Or 3. 2 Plus one. Uh huh. 3 2 Plus two. No are by squaring both sides. Uh huh. Is where would be equal to three Q whole square or three Q plus one whole square or 32 plus two whole square. Sure The square is equal to nine Q sq or 92 square Press one Press 6. Q odd. 92 square plus four plus. Welcome. All right. So now not employees. The school is equal to yeah, three times 3 q sq or three on 3 square place to Q Square plus one. Took you. I'm a government request addressed to you, press one on three times off three Q square over here to three will be coming three times of three squared plus two. You plus one or the third part. Three times off three Q square plus for Q plus one plus one. This is for three different conditions. So therefore he square would be equal to three kevin or three K two plus one on three K 3. pl one where Kevin, K two and K three. Uh some positive in teachers or something. Teachers. All right. It's gonna therefore I just change the beach today. Four. The square is equal to treat you old square or three Q plus one. Hold square on three. Topless to square. Which we are already found out by just uh just it. Is it? So the previous page, we had found it out. So therefore, hence it can be said that the square of any. Mhm. Two square, Oh, any positive in teachers is either of the former the form of three M all three emplacement. So this is the answer. Thank you friends. I hope you like the video.


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