Question

smallest positive integer in the set = {6x+15y | x,y integers}

Answer 1

21xy? I can't really understand what you mean..

Answer 2

My guess is that it is asking what the minimum value of that quantity is.

Answer 3

can you clarify?

Answer 4

Oh ok well, idk o.o sorry

Answer 5

the set is such that it contains all values described by 6x+15y where x and y are integers, for example if x=y=1 then the number would be 6(1)+15(1) = 21, so 21 is in the set, if x= -1 and y= 2 then the number would be 6(-1) + 15(2) = 24 so 24 is in the set, what is the smallest positive integer in the set?

Answer 6

smallest positive, that changes things, ok, so, let's see

Answer 7

wht have you tried so far mick?

Answer 8

i've tried "im pretty sure its 3" so far, but i cant justify it

Answer 9

so, that sounds reasonable to me, now just so I know what I can use here, what level class is this? High school, college?

Answer 10

Can 0 be an interger caus it know it means a number, so if it was 6(0) + 15(0) = 21

Answer 11

sorry i*

Answer 12

0 isn't usually considered a positive integer, it's sort of on it's own, but if it was included, that would work, (i think this question implies a>0 )

Answer 13

Very nice thought process trin

Answer 14

its a second year university number theory course

Answer 15

Hint : \(\gcd(a,b)\) is the smallest positive integer that can be written as \(ax+by\)

Answer 16

So now, mick, the first thing I'm gonna hint at is, can you factor something out of 6x +15y?

Answer 17

hmm. oh, ok hmm. and thanks. Before I go I would look up smallest positive integers and find one that fits! Im only in 6th grade and is advanced but I will try to figure it out :)

Answer 18

Keep it up trin! That was an excellent attempt

Answer 19

But 0 sounds perfect because it is the only smallest positive number and like u said sometimes it can be counted as but not usually... try 1? maybe and Thanks! Gtg for tonight

Answer 20

yeah gcd(6,15) = 3 like @ganeshie8 said

Answer 21

alright, so that gives us 3(2x+5y) right?

Answer 22

Yea, 3 is a thought!

Answer 23

+1 for the 6th grader fascinated by numbers xD

Answer 24

so now, what is the smallest you can make (2x+5y)

Answer 25

1

Answer 26

right so the smallest 2x+5y can be is 1, now we factored out that 3 remember? so we can write that as 3(2x+5y)--> 3(1) which gives you your smallest possible value :)

Answer 27

we can't minimize 3

Answer 28

we can only minimize the parentheses, and from the given we can quickly find the minimum positive value there

Answer 29

as you did, so plug chug and direct proof accomplished :)

Answer 30

They are all prime too which makes this awesome!!!!! Just a side note :D

Answer 31

ok thanks,

Answer 32

np, any questions?

Answer 33

how do we know the minimm value of 2x+5y is 1 ?

Answer 34

x=-2 y=1, minimum integer >0

Answer 35

right, how did we figure out x = -2 and y=1 ?

Answer 36

I considered that obvious, I should have proven that, good catch

Answer 37

i don't think you have to prove 1 is the smallest positive integer

Answer 38

He wants us to show how we got -2 and 1, there's a way to show that

Answer 39

interesting, so are working it without usign gcd properties.. nice :)

Answer 40

(that was my first instinct here... I like simple haha)