Question

provide proof or counterexample here:
if x and y have no common factors, then x+y is prime

Answer 1

x and y are integers

Answer 2

x = 3
y = 7

x + y = 10, not prime

Answer 3

ok lol

Answer 4

lets also suppose x and y are not prime

Answer 5

if x and y had a common factor, say r, then x = rt, y = rq
x + y = r(t + q) which isnt prime.

Answer 6

all variables are integers btw

Answer 7

but x and y share r

Answer 8

wait are you trying to do a contradiction

Answer 9

if x and y aren't prime...

x = 4
y = 45

so x and y have no common factors, and they're both composite

x + y = 49, which isn't prime

Answer 10

...i think i am lol

Answer 11

nice work you guys
im gonna come up with something harder

Answer 12

you'll see

Answer 13

i just made an evil laugh

Answer 14

lol

Answer 15

joe you still there i got one for you i think you might like
its not a proof kindof question

Answer 16

sure thing, fire away

Answer 17

Let n be an integer such that
if n is divided by 3, then the remainder is 2.
and
if n is divided by 5, then the remainder is 1.
(There can be more than one n, but give the smallest value of n that satisfies the above)

Answer 18

n is a positive integer

Answer 19

i got an answer....since i sorta kinda know about these type of problems (Chinese Remainder Theorem I believe), but I dont know the correct way to do them >.< if i turned in my solution for a test or homework, im pretty sure it would get counted wrong.

Answer 20

gonna post my solution, one sec

Answer 21

thats right to use chinese remainder thm

Answer 22

26? just a guess

Answer 23

11

Answer 24

that works, not the smallest though. really nice guess btw :)

Answer 25

yes nice guess

Answer 26

its any number congruent to 11 mod 15

Answer 27

ah yes, nice joe.

n = 3k + 2
n = 5m + 1

so 3k + 2 = 5m + 1

3k + 1 = 5m

the smallest solution of which is k = 3, m = 2

Answer 28

alright, here's my solution, beware, it butchers a lot of properties >.< my excuse is that i haven't learned this in a class yet, i was reading about it and i sotra kinda understand whats going on, but not fully >.>

Answer 29

the 15 comes from 3*5, i should have put that somewhere on there

Answer 30

\[\frac{n}{3}=Q_1+\frac{2}{3}\]
\[\frac{n}{5}=Q_2+\frac{1}{5}\]
so we have
\[n=3Q_1+2\]
and
\[n=5Q_2+1\]
\[3Q_1+2=5Q_2+1\]
\[5Q_2-3Q_1=1\]
----
\[5=3*1+2\]
\[3=2*1+1\]
----
so 3-2=1
3-(5-3)=1
-1*5+2*3=1
-1*5+(5-3)*3=1
2*5-3*3=1
so
\[Q_2=2, Q_1=3\]
but we only need one of these to find n
---
so
\[n=5Q_2+1=5*2+1=10+1=11\]

Answer 31

i use euclidean

Answer 32

then i went backwards

Answer 33

Ah, that is a much more elegant solution, is that Bezout's Identity in there?

Answer 34

im not sure
the only thing i know for sure that i used was euclidean
i dont know of bezout's identity

Answer 35

im not sure what the chinese thm says but i think it has something to do with euclidean
or gcd(5,3)=1 then we have 5x+3y=1 i think

Answer 36

its that if gcd(a, b) = 1, then there are some integers x and y such that:
ax + by = 1
it sorta looks like your 5Q_2-3Q_1 = 1

Answer 37

right i do remember it :)

Answer 38

er, rather, thats a special case, its really if the gcd (a,b) = d, then there are some integers x and y such that:
ax+by = d

Answer 39

i took 5 sense it was bigger than 3
and did
5/3
5=3*2+1
3=2*1+1
and after this i went backwards
3-2=1
3-(5-3)=1
and blah blah

Answer 40

nice nice, i like that way better than the way i learned it.

Answer 41

oops
5=3(1)+2
3=2(1)+1

Answer 42

so you are taking discrete?

Answer 43

no this was number theory lol

Answer 44

number theory*

Answer 45

ive been reading some number theory books lately, im registered to take the class this fall, but its not required to get a BS in Math, and as such, not many people at my school take it. It will be a miracle if enough people sign up for it and it doesnt get cancelled =/

Answer 46

omg you will not be dissatisfied if you take this class
i love number theory
i wish i had more of it
i'm trying to do research but it is so hard when someone is not making you do this stuff

Answer 47

i almost wasted the whole summer goofing off
i need to do it

Answer 48

It does seem extremely interesting, in one of the books there is this section on something called Diophantine (i think i spelled that right) equations, and i was totally in love lol.

Answer 49

lol
i have to get ready for bed
goodnight joe

Answer 50

goodnight, see ya later