Yes
#help-23
lean otter
x^-n
nova creek
And so what's 1/x³?
lean otter
X^-3
nova creek
And what's 2/x³?
lean otter
2^x^3?
nova creek
$\frac{2}{x^3} = 2^{x^3}$?
flat frigateBOT
Gamer Dio
nova creek
$\frac{2}{x^3} = 2x^3$?
flat frigateBOT
Gamer Dio
lean otter
2x^-3?
nova creek
Yes
lean otter
Ok
nova creek
So what's -(2/x³)?
nova creek
Yes
plush lantern
Hello, finding x for this equation is not possible, right? Teacher said all the equations given are possible.
safe radishBOT
Hello, finding x for this equation is not possible, right? Teacher said all the ...
lean otter
yeah
just multiply x by the paranthesis
and continue from there
it becomes this:
5x + 10 -3x = 2x + 14
wait wtf
thats not possible
lusty elbow
Lmaooo
plush lantern
yeah lol.. just wanted to make sure 😑😑😑
plush lantern
yup.
worthy hemlock
Could be a typo, ask your teacher
plush lantern
Wanted to make sure I wasn't doing a big silly.
Yeah will do, thanks
Solved
lean otter
i figured
worthy hemlock
Wanted to make sure I wasn't doing a big silly.
Yeah will do, thanks
Solved
If you're done with the channel, you can close it using .close
lean otter
hes still here
urban heart
Is there a map between set $\mathbb{R}^2$ and set $\mathbb{Q}$?
In other word, given an open set $A \in \mathbb{R}^2$. $A$ is open so by definition this gives a neighborhood $N_r(x)$.
How can we prove that this $N_{r}(x)$ belong to $\bigcup_{a \in \mathbb{Q}} N_r(a)$?
safe radishBOT
Is there a map between set $\mathbb{R}^2$ and set $\mathbb{Q}$?
In other word, g...
plucky elk
worthy hemlock
<:badtex:924222207251582996>
Looks perfectly fine to me \s
flat frigateBOT
soda
reef spindle
Having trouble finding the raw score with just the mean and the standard deviation
safe radishBOT
Having trouble finding the raw score with just the mean and the standard deviati...
safe radishBOT
@reef spindle Has your question been resolved?
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@reef spindle Has your question been resolved?
reef spindle
<@&286206848099549185>
reef spindle
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rugged mason
a fountain shoots water 9 feet into the air and it lands 6 feet away, what is the equation of the parabola that models this path?
safe radishBOT
a fountain shoots water 9 feet into the air and it lands 6 feet away, what is th...
@rugged mason Has your question been resolved?
rugged mason
<@&286206848099549185>
lean otter
Can you find any points belonging to the parabola?
safe radishBOT
@rugged mason Has your question been resolved?
rugged mason
no
the question was all that was given
i know the 9 is the value of k
and that's about it
rugged mason
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safe radishBOT
safe radishBOT
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broken saffron
Hello, I'm in calc 2 and need help on a problem that I'm not sure even involves any calc concepts. The background says: The included angle of the two sides of a constant equal length s of an isosceles triangle is (theta)
safe radishBOT
Hello, I'm in calc 2 and need help on a problem that I'm not sure even involves ...
broken saffron
Part A of the question asks: Show that the area of the triangle is given by A = (1/2)s^(2)sin(theta)
how am i supposed to show that exactly
safe radishBOT
@broken saffron Has your question been resolved?
broken saffron
<@&286206848099549185> hello, i was wondering if I could get help for what i believe is a geometry proofs thing
last tundra
Triangle area = 1/2 x base x height is going to help you here.
Working backwords, you can try to retrofit $ \frac{1}{2} s^2 \sin \theta $ as $\frac{1}{2}$ base x height.
flat frigateBOT
Pond
last tundra
Redraw the triangle so you have s as the base in the shape most familiar to you.
If s is the base, what's the height?
broken saffron
i apologize, give me one sec to try and think about it with your diagram
oh i think i see why now
so the height is the red line, and using sin(theta) = opp/hyp, but since we want to get the height we can cancel the "/hyp" by just multiplying by s on both sides
since the hypotenuse and base both are s
last tundra
Yes, the hypotenuse of the right-angled triangle used to define that sin ratio is s, and the line you are looking for is opp, correct.
Just be careful not to confuse the "base" of the right angled triangle used to define the sine ratio with the "base" of the triangle whose area you seek.
broken saffron
wait but then how would i actually go about writing out the proofs or whatnot
last tundra
Start with that diagram.
Pieces of information you know:
(1) The area of this triangle is 1/2 times the red line segment's length times the blue line segment's length.
(2) The blue line's length is s.
So you just need to find the red line's length.
broken saffron
so then plugging in s we have A = (1/2)bs
last tundra
However, you also know
(3) The red line segment is the side opposite theta in a right angle triangle with hypotenuse s. Therefore red line / s = so on and so on.
broken saffron
so then i could just do b = s and for height:
h/s = sin(theta), h = s*sin(theta)
then just write out A = (1/2)bh, and replace b and h with their respective values, and it would be as simple as that?
alright well thank you for the help, i really appreciate it
.close
safe radishBOT
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Available help channel!
indigo flame
safe radishBOT
indigo flame
Tips for 39pls
plucky dew
try u = sqrt(1/x -2)
plucky dew
yeah got the same thing
u forgot the dx at end tho
and rmb once u intergrated it, convert all u back to x and add constant
plucky dew
i dont think there is, usually w sqrts like these letting u as the sqrt part makes it easy
i dont think trig sub is suitable here and u cant rly change the sqrt part by factorizing or expanding
full horizon
Can someone explain to my why the group ${e, x, y, z, t, u}, *$ such that $x^2 = y^2 = z^2 = t^2 = u^2 = e$, $x * y =z$ doesn't define a group? This shouldn't be possible because a group of order 6 can be proven to have an element of order 3, but I'm unable to reach a contradiction
safe radishBOT
Can someone explain to my why the group $\{e, x, y, z, t, u\}, *$ such that $x^2...
flat frigateBOT
Yuese
full horizon
(Note that it also follows that $x * t = u$ because that's the only thing it $\textit{can}$ still equal)
flat frigateBOT
Yuese
full horizon
Ah, I think I got it, so I'll still leave the room open for someone to confirm: the contradiction is that in this case, ${e, x, y, z}$ would be a subgroup, but this can't be the case because it's order (4) isn't a divisor of the order of the original group (6) which contradicts Lagrange's theorem
flat frigateBOT
Yuese
safe radishBOT
@full horizon Has your question been resolved?
safe radishBOT
weary osprey
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safe radishBOT
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lean otter
where is my mistake?
safe radishBOT
where is my mistake?
dense coral
lean otter
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supple monolith
How would one find the value set of this function -2x^2 + 4x + 8?
supple monolith
The definition set is easy, all real values
But you could derive this function and find the value set that way to see where the derivative is zero and thats where u could find min and max
but isnt there any easier way
keen comet
but isnt there any easier way
completing the square is the other way I think about.
But differentiation sounds easier
supple monolith
yea i guess completing the square is the way to go on easier functions like this
safe radishBOT
@supple monolith Has your question been resolved?
fickle trail
safe radishBOT
fickle trail
I am trying to figure out why x + 1/2 is the same thing as 2x + 1
Cross multiplication?
white hollow
one is just a scalar factor of the other
near cave
(2x+1)/2
fickle trail
But when I plug in values such as 6 + 1/2 it doesn't make sense
white hollow
6 + 1/2 doesn't equal 13
fickle trail
oh so it's not cross multiplying
white hollow
it equals 13/2
near cave
ye
fickle trail
why are we adding a power to x?
white hollow
why are we adding a power to x?
noo I'm talking about the relevenat cubic
that you were factoring
fickle trail
oh sorry, I'm just looking at the light blue text on the right
I should have cropped that better
Lemme resend the screenshot
fickle trail
But if 12+1 is correct for the numerator.. why do I need to add the denominator? /2
if 2x + 1 doesn't need the denominator
white hollow
huh
fickle trail
I was using cross multiplication which is just wrong
I assumed that was the correct way but I don't even know if I can do that
I'm just trying to get from (x + 1/2) to (2x + 1)
white hollow
why
fickle trail
and plug in other values to do the same thing
to understand why they are the same thing
and how to get there
fickle trail
but in my example with 6 + 1/2 it doesn't work unless i add a denominator of 2 for some reason
i didn't add a denominator for 2x + 1
white hollow
they're the same thing whwen they equal 0
the y coordinate is set to 0
so you find the x coordinate
white hollow
multiply by 2
but it only works because it's 0 on the RHS
you can't exclude that
as 2x0 = 0
fickle trail
no I'm saying ignore the above, we don't know it's 0
fickle trail
Hmmm
white hollow
I don't understand
fickle trail
Ty
Does it matter if a factor includes 1/2?
Do you try to make factors whole numbers always
white hollow
it doesn't matter
iit only matters if it doesn't expand to the cubic
which is why it is sometimes necessary to multiply by a scalar
safe radishBOT
@fickle trail Has your question been resolved?
nova creek
Is the maximum less than or equal to every element in the set?
Yeah my bad
But it is ≥ every other element right?
Is there any smaller element that's also ≥ every other element?
Then it's the supremum
In the field of reals, yes
In the field of rationals, the supremum doesn't exist
Well, the supremum is a number
So if the number isn't in the field, it doesn't exist
99.99% of the time though the field is the reals
So yeah, the supremum would be ³√29
Yep
safe radishBOT
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Available help channel!
cobalt flint
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cobalt flint
hello I feel like somethings a bit off with my answers
can someone let me know if they’re wrong and some point in the correct direction
hollow oak
3y²-3 for fy
hollow oak
For fyy it will be 6y
hollow oak
Yeah so in the next step it's just normal differentiation keeping the other variable constant
cobalt flint
what about fxy?
hollow oak
Fyx would be 0
hollow oak
0
lethal scaffold
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safe radishBOT
lethal scaffold
.reopen
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neat kiln
Easy just place x in the denominator and you get vertical asymptote x,=0
safe radishBOT
@lethal scaffold Has your question been resolved?
lethal scaffold
Easy just place x in the denominator and you get vertical asymptote x,=0
well the question is requiring to make a function then describe certain things about it
like this
or this
safe radishBOT
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lucid atlas
safe radishBOT
lucid atlas
I’ve gotten to this point
But I’m not sure how to write it in terms of an inequality
I searched online and it said x<-3/4
But when I put it into original equality I got 0 so I’m not sure how that works
proper crypt
lucid atlas
How would I do this without doing all that eg in an exam
sleek plank
you could use quadratic formula
lucid atlas
Right and it would get me x =-3/4 and x=2 right?
How would I put this into the inequality form
Online says x<-3/4 x>2
toxic stratus
you could use quadratic formula
factorising is more appropriate here
sleek plank
plug in 0, 0 and see if it satisfies the inequality
lucid atlas
Online says x<-3/4 x>2
How do I get this?
toxic stratus
graph the parabola
and see when it is above the x axis
this will tell you the values for which it is >0
lucid atlas
Is there any other way to do it?
toxic stratus
you have already calculated the intercepts
you can plug in values in each interval that the intercepts split the real line into
but that's is more tedious than simply graphing
you already know the intercepts presumably from solving the quadratic =0
or perhaps just factoring
toxic stratus
so just draw a rough shape of the parabola
and determine when it is above the x axis
x axis
not y
sleek plank
if it's shaded inside or outside the parabola
toxic stratus
shading?
lucid atlas
Is that how you would do it in an exam?
sleek plank
that's how i would
lucid atlas
Is there any other way to do it instead of graphing it
sleek plank
instead of graphing it, you can just plug in the coordinates 0, 0 to see what part you shade
inside or outside
sleek plank
take the example above
if you plug x = 0 and y = 0 in the inequality, it wouldn't become true
after you evaluated it
toxic stratus
as in something like this
no that's not really what we're looking for
the original inequality has no y in it
we are not concerned about the y values
sleek plank
y is only there so it shows up
sleek plank
desmos doesn't graph it without the y being there
sleek plank
it's just an example as to what shaded parabolas look like
lucid atlas
I’ve got the two x values, now how do I put it into the inequality form
toxic stratus
you don't shade parabolas for these inequalities
the parabola tells you whether the values are greater than or less than 0
lucid atlas
I also know what the inequalities are I just don’t understand how to get that answer
toxic stratus
when it's above the x axis then the quadratic is >0
lucid atlas
X<-3/4 x>2
toxic stratus
yes
lucid atlas
Yes but how can I do this without using the parabola
toxic stratus
you should draw the parabola
lucid atlas
Is that the only way?
sleek plank
Yes but how can I do this without using the parabola
think about it like 2 < x < -3/4
toxic stratus
that the intercepts split the real line into
but it is more tedious as i have said
you want to draw the parabola
lucid atlas
think about it like 2 < x < -3/4
Right okay, then how does that become x>2
toxic stratus
you know the intercepts
so you know when the quadratic changes signs
i.e when it switches from being <0 to >0
so you just have to check whether it's >0 or <0
you do this by plugging in a value in the interval you want to check
so say for example you plug in x=0
then you just get -6
so you know that on the interval -3/4 < x < 2 the quadratic is negative
so that means outside that interval it'll be non-negative
lucid atlas
Right sorry i understand partially up to the last two parts
If I plug in x=2
I get 0
safe radishBOT
@lucid atlas Has your question been resolved?
lucid atlas
Okay this whole thing confused me
I’ve got it now but without the plotting on the graph thing
I did it with a bit of trial and error
Thank you very much for your time anyway!
.close
bright turtle
I don’t understand how this got 9.33?
safe radishBOT
I don’t understand how this got 9.33?
bright turtle
Help pls
sleek plank
6/18 simplifies to 1/3 because both 6 and 18 are divisible by 6
when you divide both 6 and 18 by 3 you get 1 and 3 respectively
keen agate
, my teacher explained a identical question on the board but i learned nothing from it, im supposed to expect a question like so on the test
safe radishBOT
, my teacher explained a identical question on the board but i learned nothing f...
sleek plank
the trajectory of the bounce angle will be equal to the 90 - the angle of impact
ur supposed to use trigonometry to find this angle
and then subtract what you got from 90
then you subtract THAT by 180
and draw a line at that angle
safe radishBOT
@keen agate Has your question been resolved?
safe radishBOT
foggy salmon
hm
fickle trail
.close
Sorry, using iPad and Discord, it wants to create a new channel when I add screenshot
.close
safe radishBOT
Available help channel!
opaque sorrel
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opaque sorrel
hi can someone help me with this?
i did part a i and ii
but im not sure how to do b
my idea is to do by cases, like
cases where all letters are distinct + cases when there are 2 Es or 2Ls + cases where there are 2 Es and 2Ls + cases where there are 3 Es
merry sleet
you are talking about iii?
opaque sorrel
yes
opaque sorrel
gets you to the result quickly
so itll go like
(6C4 * 4!) + 2 * [(3C2 * 5C2 * 3!) / 2!] + [3C2(select Es) * 4!] / 2!*2! + [6C1 * 2!]
merry sleet
i'm not sure about all of them
foggy salmon
its u
opaque sorrel
hi
foggy salmon
hi
opaque sorrel
i'm not sure about all of them
\
merry sleet
last one i would have had 5c1 for the choice of the letter that isnt E and also 4C1 for its place inside the codeword
merry sleet
i dont get what you mean
opaque sorrel
oh like
so for the last one
when there are 3 Es
i selected the last one with 5c1
wait
oh it should be 6c1
wait
merry sleet
it cant be E
opaque sorrel
but theres alr 3 Es
J E W L R Y
E L
E
i selected all the Es so i just counted everything else
merry sleet
yeah so there are 5 possible letters remaining
wer'e not doing probabilities
where each letter has equal chance of being drawn
so its 5C1
merry sleet
why by 2! ?
merry sleet
there are 4 spots for the non E letter
opaque sorrel
4!/3!
opaque sorrel
ahhhh
merry sleet
yeah its not the end
opaque sorrel
oh wut
opaque sorrel
oh..
merry sleet
and im lazy
opaque sorrel
yes
opaque sorrel
would it work if i choose the Es first
wait
oh wait
i dont have to
cuz theyre all the same
merry sleet
its just about knowing their place
opaque sorrel
4!/2!2!
merry sleet
chose 2 spots among 4
merry sleet
xor*
merry sleet
you have to chose their spots though
opaque sorrel
(6C4 * 4!) + 2 * [(3C2 * 5C2 * 3!) / 2!]
ill permute them later
hmm
so i need to choose others
so
merry sleet
(6C4 * 4!) + 2 * [(3C2 * 5C2 * 3!) / 2!]
you didnt get rid of anything
merry sleet
do it from the beginning i think
merry sleet
yeah
opaque sorrel
then uh
merry sleet
then you have chosen all the letters
opaque sorrel
ah wait
merry sleet
there are 4 spots in your word
merry sleet
but the Es are equivalent
opaque sorrel
hmm
the EEE is wrong
its 6C1 * 4 = 24
right
ah wait
bruh
5C1
bruh
bruh
merry sleet
got ideas for b?
merry sleet
lets place the two parents first
opaque sorrel
total number of ways when mr lee sits with mrs lee - number of ways mr lee sits with mrs lee and the children are all sat together
merry sleet
why -?
merry sleet
only the children right?
opaque sorrel
yyea
merry sleet
it would need to be
total number of ways when mr lee sits with mrs lee - number of ways (mr lee sits with mrs lee are sat together) and (the 3 children are sat together)
opaque sorrel
oh
LOL
yea
i did
oopsie
lets do it
so total number of wyas with mr lee and mrs lee sitting toegther
its uh
so i fix the parents
then just permute the rest
so for the rest it would be 8!
with parents 9!
then because its a circle
so (9-1)!
so 8!
merry sleet
they are 8 in total
opaque sorrel
wut
merry sleet
oh nvm
opaque sorrel
lmao
opaque sorrel
wait so 8! is correct?
opaque sorrel
heh
oh right
then for the
parents together and children are sat together case
its uh
parents is one group
children is one group
so 7! in total
but circle so 6!
merry sleet
dont be confused
merry sleet
the children themsleves dont form a circle
they are separated by the parents
that are sat together
opaque sorrel
wait wat
separated by parents?
i thought im finding if the children are sat together
merry sleet
opaque sorrel
yea
merry sleet
in both cases the parents are sat together
opaque sorrel
yes
merry sleet
so you treat the childrens not as a circle
opaque sorrel
i didnt tho
merry sleet
but circle so 6!
.
merry sleet
everything?
merry sleet
okay
opaque sorrel
so there are 5 other circlees
so 5!
then there are two "groups"
children and parents
so 7!
then because its a circle
6!
merry sleet
oh ok
merry sleet
its right then
merry sleet
yeah
merry sleet
how many was that?
opaque sorrel
72000
merry sleet
yeah they got plenty of room
merry sleet
this kind of thing is always tedious
merry sleet
sadly this isnt recursive
untold wadi
Hello I was wondering if someone could verify if this problem is correct?
untold wadi
This is probability and I worked on it with a friend but am not sure if it's correct or not
unreal dock
it looks right to me
untold wadi
.reopen
safe radishBOT
✅
untold wadi
it looks right to me
Okay
unreal dock
🫡
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Available help channel!
dense compass
Can someone explain why it goes from 144 to 169?
safe radishBOT
Can someone explain why it goes from 144 to 169?
dense compass
Arent you supposed to set it equal to 0?
hot thistle
set what to 0
dense compass
the right side
hot thistle
no
dense compass
do you know what that is?
then apparently not
hot thistle
do you know the equation for a circle?
dense compass
x^2 + y^2 + 2gx + 2fy + c = 0
hot thistle
or more common is $(x-h)^2 + (y-k)^2 = r^2$
flat frigateBOT
maximo
hot thistle
actually it may not be more common but it’s more helpful here
anyhow, we want to get the equation you’re given into that form
to do so we want to get (x-something)^2
since we have x^2 - 6x, that something has to be a 3
since (x-3)^2 = x^2 - 6x + 9
so it’ll look like this:
x^2 - 6x =
x^2 - 6x + 0 =
x^2 - 6x + (9 - 9) =
x^2 - 6x + 9 - 9 =
(x-3)^2 - 9
do you understand how i got here so far
dense compass
since (x-3)^2 = x^2 - 6x + 9
Where does the 6x go here? I can see the x^2 becomes x and the 9 becomes 3. But what about the 6x?
think thats the part im stuck on
hot thistle
(a+b)^2 = a^2 + 2ab + b^2
(a+b)^2 =/= a^2 + b^2
sometimes it works but far from always
hot thistle
nono
you can’t
that’s the point
so we get (x-3)^2 = x^2 - 6x + 9
but we start with only x^2 - 6x
do we add 9 and subtract 9
dense compass
we + (9 - 9)
dense compass
Ok I think im starting to understand
I just need to look at it more
wait @hot thistle why are we adding the 9's? I get its converting x to square form, but why 9 specifically?
hot thistle
it’s the number that lets us write (x-something)^2
for example if we had x^2 + 2x
we want to add 1
so we get (x+1)^2 - 1
hot thistle
well we want a number so that when we expand it out we get -6x
so since (x-a)^2 = x^2 -2ax + a^2
we want -6x = -2ax
so a = 3
so we add and subtract a^2=9
dense compass
ok its like 12am so ill have another crack at it in the morning, getting close though. Thank you!
.close
trim jackal
For 7 part (g), I believe the statement is true. If I was going to prove that it was true, could I use a contrapositive or maybe a contradiction?
safe radishBOT
For 7 part (g), I believe the statement is true. If I was going to prove that it...
tiny wraith
the biggest number in S(x) is always x
trim jackal
Right
tiny wraith
so if S(a) = S(b) their biggest numbers would also be equal, and thus a = b
trim jackal
In terms of a contrapositive, right?
tiny wraith
you could say that if a weren't equal to b and S(a) did equal S(b) that would imply that a = b, so the statement that a is not equal to b and S(a) is equal to S(b) could never be true
lean otter
hello little boys
safe radishBOT
hello little boys
lean otter
how do i move y^2 to the other side and x also
to get y^2=ax
i mean like whats the metho to rearrange it
it just shows they rearranging it
lilac reef
multiply both side by 40?
lean otter
okay is that the only method
stray bluff
thats the easiest
lean otter
so if it was 40 in the original equation i would multiply both sides by 1/40?
lilac reef
yes
stray bluff
yea or just divide by 40
lean otter
that would be 40/1 times 1/40 and how does that work
like i dont understand the logic behind it could u explain pls
multiplYING 2 fractions
lilac reef
40/1 * 1/40 = (40 * 1) / (1 * 40) = 40/40 = 1
sorry, i forgotten latex already, so I can't give proper format
lean otter
when multiplying fractions do i cross multiply?
wait
oh
do i
when multiplying 2 fractions do i cross multiply or multiply straight through
multiplying numberators by eachother
and denominators by achother
lilac reef
I am sorry, I am not a native english speaker
lean otter
ok i will send photo
lilac reef
I am not too familiar with the terms cross or straight multiplication
lilac reef
ok i will send photo
sure that'll help
lilac reef
yeah
lean otter
when do u cross multiply
because thats something i remember but dont know when to use it
lilac reef
i am not too sure... I assume if you cross multiply, you divide
lilac reef
thank u for helping me
you're welcome, if you do not have anymore question, you can close the channel by ".close"
lean otter
.close
safe radishBOT
@timber briar Has your question been resolved?
safe radishBOT
@timber briar Has your question been resolved?
safe radishBOT
@timber briar Has your question been resolved?
lean otter
Where have you seen that cube before?
peak estuary
I'm just gonna throw the words menger sponge into this thread although I doubt you mean those
void plank
Something about this reminds me of fractals tho I don't know what
void plank
This looks very similar to a merger sponge
pliant inlet
safe radishBOT
pliant inlet
b) onwards
i have a strange solution
no clue how to proceed
proofs are not my strong suit
foggy salmon
hm factorised u get
2x(x+2)(x+1)
yea?
2x(x+1)(x+2)
imma reorder that
so for any arbitrary int a
like if we take a mod 3 we would get either 0,1,2
pliant inlet
yeah
foggy salmon
we can use case by case
if its 0 then its div by 3
if its 1 then a+2 is div by 3
if its 2 then a+1 is div by 3
n that covers all the cases
pliant inlet
like if we take a mod 3 we would get either 0,1,2
can you explain this bit?
foggy salmon
yes
um
like cuz
if u take mod n of some int
ull get stuff from
0,1,2,..., n-1
n would reduce back to 0
like a cycle
so in this case mod 3 will just be
0,1,2
lean otter
i don't think they are supposed to use mod to explain the proposition
foggy salmon
oh?
pliant inlet
i dont remember using mod in the textbook
grave flint
poopy mod
foggy salmon
u can just do the same thing
but with
a = 3k
or
a = 3k + 1
or
a = 3k + 2
for some k
pliant inlet
right
foggy salmon
for some integer k
pliant inlet
mhm
foggy salmon
yea
pliant inlet
so i can start with
foggy salmon
then
foggy salmon
if its 0 then its div by 3
smt like this from there
pliant inlet
a = 3k, a= 3k + 1 a = 3k+2
grave flint
each is a case
foggy salmon
umm
well
like
if
a = 3k
then a is div by 3
if
a=3k+1
then a+2 = 3k+1+2
=3(k+1)
and k+1 is an int
thus a+2 is div by 3
last case,
if a=3k+2
then
a+1=3k+2+1
pliant inlet
a + 1 3k + 1
pliant inlet
right...
foggy salmon
and those r the 3 cases
foggy salmon
essentially the same as mod lol
foggy salmon
um
i mean
it was kinda
intuitive
that 1 of em wld b divisible
so just add some dummy var to
um
idk
beef it up
pliant inlet
yeah but proving it is hard lmao
pliant inlet
alright
foggy salmon
complete explaination anw
pliant inlet
or d
foggy salmon
its similar to the b
where u got
a,a+1 and a+2
so 1 of em wld div by 3
so therell b nth in the denom
n itll div n give u an int
from part b
and as x in N
when div 3 u get some int >=0
then
multiplication is preserved in N
pliant inlet
multiplication is preserved in N
i got most of that until this bit
foggy salmon
closed under?
pliant inlet
right closed under multiplication