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[2Opts] Prove each of the following: (a) If alb and a/c; then af(b _ c) (6) a =b (mod m) ^ c= d (mod m) = a+c=b+d (mod m)...

Question

[2Opts] Prove each of the following: (a) If alb and a/c; then af(b _ c) (6) a =b (mod m) ^ c= d (mod m) = a+c=b+d (mod m)

[2Opts] Prove each of the following: (a) If alb and a/c; then af(b _ c) (6) a =b (mod m) ^ c= d (mod m) = a+c=b+d (mod m)



Answers

Write complete proofs for each of the following exercises. Given: B-A-I; $\mathrm{RN}=\mathrm{RI}$ and $\mathrm{AN}=\mathrm{AI}$ Prove: $\quad \mathrm{BR}+\mathrm{RN}>\mathrm{BA}+\mathrm{AN}$

In this question it is given that maybe the AM of BNC and B. B the gm of C N A. Then we have to prove that sees the H. M of A. N B. Given okay, have a good time gm of okay, Mhm CN e is equal to be yeah Gm of C N A. Is the square root of C into E. Is equal to be squaring both side in to see is equal to be square. Consider this equation one given I am of me and see is equal to E am of BNC is deep lists. See upon two is equal to B. B plus C Is equal to two. Mm Multiply both side. Mhm Bye B. We get mhm. Be into B plus C. is equal to two into and to be be square place being to see is equal to two into A into me. People from question one, it's attitude be square is equal to A into C. Plus being to see is equal to two and 2 A and two B. From I think equation one. Mhm Taking common see A plus B Is equal to two into a into B. Therefore C is equal to to into into me a phone A Plus we therefore see is hmm. Off E and B approved. Mhm

You know this problem? We're wanting to write a proof. Do this. Let's attract one on each side, which we can do as long as we do it to both sides. Here on the left. Getting and common denominator, We have a minus. Pay will be minus B. Overbey on the right. We have C over D minus D over D. You're just getting a common denominator. So this is a minus. Bi Overbey. It will c minus D over d, just like we want.

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.

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.


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