okay LET'S DO SOME MATH FOR FUN and here i have this question for you guys and this is
from the Harvard MIT math competition we are going to find the smallest positive integer n
so that this expression is an integer WOW because we have a square root here inside of a square root
and we have another one we actually have done a video on the cube root situation like this
that expression gave us an integer so we have a pretty good hope for this right
anyway as always please pause the video and try this first
okay you did it good if you did that's really good and let me know okay let's see how we can
do it so perhaps typically we can just call this to be let's say k that can equal this
and perhaps let me just indicate that k is an integer all right because we have square roots so
maybe it's a good idea to just square both sides so let's go ahead and do so
right here and uh we are getting k squared is equal to this thing squared which is just 100
plus square root of n plus two times this and that which is just two square roots and then
we have the inside times inside which will give us a hundred square which is one thousand i mean
ten thousand and then minus n and then we add this thing squared which is 100 minus square root of n
and this is actually pretty nice because this and that cancel and we can combine the hundreds
so k squared is equal to 200 plus 2 times the square root of 10 000 minus n so it looks pretty
easy right now right and let me just show you guys what i did at first because i thought it was just
that easy because this is the only problem that we have huh because if this square root gives us an
integer then the hosting is an integer so this is what i did notice that i'm not going to choose n
is equal to 0 because n has to be positive but if you look at that this is just 100 squared
so i make this i set this to be square root of 99 squared because we are taking away
you know n from there well 99 squared is the same as 9 8 0 1. so i'm just going to do 10 000
minus n and set it equal 9801 and solve this and we get n is equal to 199
dandiyo that's it then you get some no no no no no no no this is wrong take a look if
you put this back to the original expression unfortunately it doesn't give us an integer
yeah that's bad sorry and let's talk about why if we put this back here
yes we just get 19 square instead of the square root so we have 99 times 2 and then plus 200.
that's 490 something no 398 yeah unfortunately that is not a perfect square because you see we
have k squared is equal to this so we do have to pay attention to the left hand side so now we have
to make more observations the first one is the following k squared is equal to 200 plus two times
that we can factor out two so this right here will tell us k squared has to be even and because k
squared is even this right here implies and it's actually like a bi-direction if you would like
k is even okay so just keep that in mind okay now the other thing that we have to talk about
is what if n was just equal to zero then that was the easiest case right because we just get square
root of 100 plus square root of 100 and that's going to give us 20. so that would be 20 and if
we square both sides then we know we will just get 20 squared here yeah but we don't get 20 squared
yeah we don't get 20 squared why because again k i mean n has to be positive now
the biggest this right here could be was 20 squared what k is equal to 20
but we know k is even and we're taking away a little bit
so the next thing that we can get is in fact we have to work out when k is equal to 18.
all right again the biggest right here the biggest and technically this is not achievable because
n cannot be equal to zero so the biggest was uh 20 squared but that's when
n is equal to zero but this is not achievable so we go to the next one the next even
well the previous even number right so 18 that's our consideration so we are
just going to make this equal to 18 and then we just plug that back so we have 18 squared
so 18 squared and that's equal to 200 plus 2 square root of
yeah just solve this equation now and that's pretty much it so let's see 18 squared is 224
minus 200 is 124 and divided by 2 is 62 and then we make that equal to the square root
10 000 minus n and of course we can just do the usual business square both sides and then work
this out right and i'll just tell you guys that n is equal to six thousand one hundred fifty six
seriously you are asking for the smallest integer n and then the smallest answer is 6156 ready
yeah really because if you put this back there yes we get our integer yep yep if you like solving
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for sponsoring this video so now i have another question for you guys is it possible for us to
find the biggest negative integer n so that the same expression gives us an integer and we notice
when n is negative this right here is complex so what do you think i'm gonna do this to you guys
wow