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Find a pair of positive integers a <b such that a 2 + b 2 = 223074.and b...

Question

Find a pair of positive integers a <b such that a 2 + b 2 = 223074.and b

Find a pair of positive integers a <b such that a 2 + b 2 = 223074. and b



Answers

Find two numbers $a$ and $b$ such that $a<b$ and $|a|>|b|$

Were asked to show that if a and B are too positive integers then to to the A minus one mod to to the B minus one is equal to to to the a mod being minus one. Now, by the division algorithm, we have that assuming a is greater than be without loss of generality is going to be be times the floor of a over B. Yeah, plus a mod be by the division algorithm. And then we have that to the A minus two the A might be This is equal to to to the b floor of a ever be plus a mod be minus two to the a mod be. We can factor out the two to the Aamodt be from both of these to get to to the A might be times to to the b times the floor of a over B minus one. Now we have that to the B is congruent toe one model to the B minus one. This is a result which was approved earlier. This is actually just the fact that this is the same number. So we have that to to the b times, the floor, they ever be is congruence toe one mod to to the B minus one. Since this is a multiple of to to be, it's another way of saying this is this is congruent 21 to the floor they might be, which is, of course, congregant Toe one mod to to the B minus one. And therefore it follows that we also have to to the B times the floor of a over B minus one is congruent 20 mod to to the B minus one. So this implies that to to the A minus two to be a model B you saw earlier this was congruent to to to the a mod be times due to the B times the floor of a or B minus one. This is congruent to zero. So this is going to be concluded 20 mod to to the B minus one and adding to to be a mod be to both sides. We get that to to the A is congruent to to to the a Barbie mod to to the B minus one. Now subtracting one side from both sides. Okay, we get to to the A minus one is congruent to to to the a ma be minus one mod to to the B minus one and since is congruent to be mod m is equivalent with a modern m equals B madam, It follows that to to the A minus one. Maude to to the B minus one. Yeah, is equal to to to the a mod be minus one mod to to the B minus one. And we see that this is the same as to to the a mod be minus one. Mhm. And so we have shown that to ta minus one. Mod to to the B minus one is equal to to to the a mod be minus one.

Okay, So, first statement, we want to show that if ain't divide B and B divides a then a easy to be or eggs even too negative. So to show this, if we have eight divides, quite be. This implies there exists an Internet sea in indigenous such that a c is it? It would be an opposite when eight would certainly be devised. A. This imply that their existence interject d images such that B d is equal to a So what we'll do is we'll substitute Well, sorry we will substitute be into here So we have a sea turns my deep is it to a expanding? We have a C D is equal to a But you notice that these two A's us are equal So they're the same letter. So that means by the identity property of multiplication CD must be equal to one. So this imply that CD is even to one. But since Cmdr. Vintage is by definition, we can't have any fractions. So we are left with. Either See is even to be easy to one or C is a city is a desert months one, because see India in stitches. Now what we'll do is we'll substitute this form into this one. First up. So we have seasons into one so easy to be so eight times one is one. And it substituted this one into the same one. Here we have 800 miles. One. So that's negative, eh? Is it? It should be or a easy to negative, and we've just proven their results.

All right, a little bit about they want to slug in 18 and 10 for A and B. And make sure we get to that answer that they get to. And I'm gonna go ahead and do it like this. 18 is nine times to 10 is two times five. And so I'm immediately gonna see this. Nine is a square is three time streets. I can pull three out two times. Two means I can pull it to out. Leaving the five behind. Three times two is six. And so indeed, yes, this is true. Six were five Happens is 18 and B is 10.

Some vectors we have to find the products.


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