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FoolForMath
Group Title
Fool's problems of the day,
[** EDIT: Complimentary problem added **]
On the April fools' day I give you three cute problems on number theory/Combinatorics:
\((1) \) Find the number of nonnegative integral solutions of \(3x+4y=120 \)
[Solved: @2bornot2b ]
\( (2) \) If \(81x + 64y = n\) find the greatest possible of \( n \) such that both \( (x, y) \) are not positive.
\( (3) \) Let \(n\) and \( m\) be positive integers. An \( n \times m \) rectangle is tiled with unit squares. Let \(r(n,m) \) denote the number of rectangles formed by the edge of these unit squares. Thus, for example, \(r(2, 1) = 3\). Can you find \( r(11, 12) \)?.
[ Solved by @satellite73]
PS: The problems are arranged in an increasing order of difficulty. (However, (2) could seem very very easy to a number theorist!)
** Complimentary for those who feels these are very hard:**
If the tangent the to the curve \(\sqrt{x} + \sqrt{y} = \sqrt{a} \) at any point on cuts xaxis and yaxis at two distinct points. Can you find the sum of the intercepts ?
Good luck!
 2 years ago
 2 years ago
FoolForMath Group Title
Fool's problems of the day, [** EDIT: Complimentary problem added **] On the April fools' day I give you three cute problems on number theory/Combinatorics: \((1) \) Find the number of nonnegative integral solutions of \(3x+4y=120 \) [Solved: @2bornot2b ] \( (2) \) If \(81x + 64y = n\) find the greatest possible of \( n \) such that both \( (x, y) \) are not positive. \( (3) \) Let \(n\) and \( m\) be positive integers. An \( n \times m \) rectangle is tiled with unit squares. Let \(r(n,m) \) denote the number of rectangles formed by the edge of these unit squares. Thus, for example, \(r(2, 1) = 3\). Can you find \( r(11, 12) \)?. [ Solved by @satellite73] PS: The problems are arranged in an increasing order of difficulty. (However, (2) could seem very very easy to a number theorist!) ** Complimentary for those who feels these are very hard:** If the tangent the to the curve \(\sqrt{x} + \sqrt{y} = \sqrt{a} \) at any point on cuts xaxis and yaxis at two distinct points. Can you find the sum of the intercepts ? Good luck!
 2 years ago
 2 years ago

This Question is Closed

Ishaan94 Group TitleBest ResponseYou've already chosen the best response.1
dw:1333293300537:dw
 2 years ago

Ishaan94 Group TitleBest ResponseYou've already chosen the best response.1
Why did everyone leave? :(
 2 years ago

Ishaan94 Group TitleBest ResponseYou've already chosen the best response.1
Now how do I get the number of positive integral solutions? I don't wanna use hit and trial method :/ We know \(y\le30\) and \(x\le40\).
 2 years ago

FoolForMath Group TitleBest ResponseYou've already chosen the best response.10
HINT: Linear Diophantine equation.
 2 years ago

lgbasallote Group TitleBest ResponseYou've already chosen the best response.0
these problems would be unsolveable and fool will greet us April Fool's Day mmhmm <nods> that would be epic
 2 years ago

FoolForMath Group TitleBest ResponseYou've already chosen the best response.10
(1) is very much solvable by almost anybody.
 2 years ago

experimentX Group TitleBest ResponseYou've already chosen the best response.0
both of \( (x, y) \) are not positive??? either x is negative or y is negative or both are negative?
 2 years ago

experimentX Group TitleBest ResponseYou've already chosen the best response.0
looks like n is greatest common divisor
 2 years ago

Ishaan94 Group TitleBest ResponseYou've already chosen the best response.1
Diophantine sounds like organic chemistry. I'm googling it. 10?
 2 years ago

FoolForMath Group TitleBest ResponseYou've already chosen the best response.10
Lol, he (Diophantus) is the sometimes called the father of algebra.
 2 years ago

Ishaan94 Group TitleBest ResponseYou've already chosen the best response.1
It took me time but the Linear Diophantine thing is really nice. Thanks.
 2 years ago

experimentX Group TitleBest ResponseYou've already chosen the best response.0
is the answer of 2 1??
 2 years ago

FoolForMath Group TitleBest ResponseYou've already chosen the best response.10
1 is the not the right answer.
 2 years ago

experimentX Group TitleBest ResponseYou've already chosen the best response.0
must be infinity then
 2 years ago

FoolForMath Group TitleBest ResponseYou've already chosen the best response.10
No it's not infinity.
 2 years ago

2bornot2b Group TitleBest ResponseYou've already chosen the best response.2
There are 11 non negative integral solutions to the given diophantine equation
 2 years ago

2bornot2b Group TitleBest ResponseYou've already chosen the best response.2
Am I right?
 2 years ago

FoolForMath Group TitleBest ResponseYou've already chosen the best response.10
Well done @2bornot2b!
 2 years ago

FoolForMath Group TitleBest ResponseYou've already chosen the best response.10
If other want, you may like to the post the solution too.
 2 years ago

2bornot2b Group TitleBest ResponseYou've already chosen the best response.2
Using euclidean reduction it can be shown that the integral solutions are of the form \(x=4n\) and \(y=3(n10)\) where n is any integer. One can also use the formula to solve diophantine equation or any other method. Now for positive n, 4n is always positive, so we can't take into account any negative n. Again the expression (n10) must be negative so as to have positive y, which is possible for n=0,1,2,3....,10. So we have 11 solutions
 2 years ago

2bornot2b Group TitleBest ResponseYou've already chosen the best response.2
And yes for n10=0, the value is true too.
 2 years ago

Ishaan94 Group TitleBest ResponseYou've already chosen the best response.1
For 2nd 81*64?
 2 years ago

FoolForMath Group TitleBest ResponseYou've already chosen the best response.10
@Ishaan94: Ishaan you are very close.
 2 years ago

Ishaan94 Group TitleBest ResponseYou've already chosen the best response.1
81*64  1? lol
 2 years ago

FoolForMath Group TitleBest ResponseYou've already chosen the best response.10
haha, nopes :)
 2 years ago

experimentX Group TitleBest ResponseYou've already chosen the best response.0
i was thinking the same .. lol
 2 years ago

experimentX Group TitleBest ResponseYou've already chosen the best response.0
but it seems 0 is nowhere close to 81*64
 2 years ago

Mr.Math Group TitleBest ResponseYou've already chosen the best response.0
Isn't it obvious that if both \(x\) and \(y\) are nonpositive, then the largest possible value of \(n\) is \(0\)?
 2 years ago

Mr.Math Group TitleBest ResponseYou've already chosen the best response.0
I'm talking about the second problem.
 2 years ago

experimentX Group TitleBest ResponseYou've already chosen the best response.0
yeah ... but both \( (x, y) \) are not positive. might mean both cannot be ++, but can be + or + ... just considering possibility
 2 years ago

Ishaan94 Group TitleBest ResponseYou've already chosen the best response.1
It is, but at first I presumed it was both x and y can't be nonpositive integral :/
 2 years ago

Ishaan94 Group TitleBest ResponseYou've already chosen the best response.1
If you'll read the first question you might understand the cause of my presumption :/
 2 years ago

experimentX Group TitleBest ResponseYou've already chosen the best response.0
still, considering this http://en.wikipedia.org/wiki/Diophantine_equation#Linear_Diophantine_equations i still think 0 is the answer, since 81and 64 do not have any common divisor
 2 years ago

Mr.Math Group TitleBest ResponseYou've already chosen the best response.0
Oh he probably meant "such that x and y are not both positive", meaning that only one of them can be positive.
 2 years ago

experimentX Group TitleBest ResponseYou've already chosen the best response.0
in that case .. Mr. Math must be right.
 2 years ago

Ishaan94 Group TitleBest ResponseYou've already chosen the best response.1
We need to minimize the indicated portion, which is only possible if \(n\le 0\), and 0 is the largest as the rest of the values are negative. dw:1333297533937:dw
 2 years ago

Mr.Math Group TitleBest ResponseYou've already chosen the best response.0
The answer of (2) is \(0\), unless something's wrong with the question.
 2 years ago

Ishaan94 Group TitleBest ResponseYou've already chosen the best response.1
FoolForMath for the 3rd question, Can you tell the value of r(3,2)?
 2 years ago

experimentX Group TitleBest ResponseYou've already chosen the best response.0
@FoolForMath what do you say??
 2 years ago

Ishaan94 Group TitleBest ResponseYou've already chosen the best response.1
FoolForMath WHAT SAY YOU? *This is more cinematic hehe*
 2 years ago

Mr.Math Group TitleBest ResponseYou've already chosen the best response.0
What does "is tiled with" mean? (My English is failing me)!
 2 years ago

Ishaan94 Group TitleBest ResponseYou've already chosen the best response.1
Hmm made up of, maybe
 2 years ago

Mr.Math Group TitleBest ResponseYou've already chosen the best response.0
But why is \(r(2,1)=3\)?
 2 years ago

FoolForMath Group TitleBest ResponseYou've already chosen the best response.10
Ishaan, \( r(3,2)=18 \).
 2 years ago

satellite73 Group TitleBest ResponseYou've already chosen the best response.0
no you are trying to find n is r(3,2)=16?
 2 years ago

Mr.Math Group TitleBest ResponseYou've already chosen the best response.0
@satellite, if both \(x\) and \(y\) are not positive then obviously the maximum value of \(n\) is \(0\). If only one of them has to be nonpositive, then we can choose \(x>>> 0\ge y\) and then n would have no maximum.
 2 years ago

experimentX Group TitleBest ResponseYou've already chosen the best response.0
number of rectangles?? you mean there are lot's of rectangle???
 2 years ago

satellite73 Group TitleBest ResponseYou've already chosen the best response.0
not both postive ok i see r(3,2)=18
 2 years ago

FoolForMath Group TitleBest ResponseYou've already chosen the best response.10
I think both \( x \) and \( y\) are not positive means either of them can be positive and both of them can be negative. x=1, y=1 then, 8164 = 17 you have to maximize this value.
 2 years ago

satellite73 Group TitleBest ResponseYou've already chosen the best response.0
?? lost me on that one
 2 years ago

experimentX Group TitleBest ResponseYou've already chosen the best response.0
x=inf, y=1, n = inf
 2 years ago

satellite73 Group TitleBest ResponseYou've already chosen the best response.0
since gcd(81,64)=1 we can solve \[81x+64y=1\] for x and y and so can solve \[81x+64y=n\] for any n
 2 years ago

Ishaan94 Group TitleBest ResponseYou've already chosen the best response.1
But as long as you have positive y intercept and x intercept, you will always have positive solutions for (x,y) subject to \(x,y \in \mathbb{R}\).
 2 years ago

FoolForMath Group TitleBest ResponseYou've already chosen the best response.10
sat, if x and y are nonnegative integers, then the the greatest integer that cannot be written in the form ax+by is ab − a − b. [assuming (a,b)=1 ]
 2 years ago

experimentX Group TitleBest ResponseYou've already chosen the best response.0
x=inf, y=1, n = inf what about this ??
 2 years ago

satellite73 Group TitleBest ResponseYou've already chosen the best response.0
i have an answer i like for the second to last one
 2 years ago

satellite73 Group TitleBest ResponseYou've already chosen the best response.0
whew. left the house, had a beer and thought about it a little more clearly
 2 years ago

FoolForMath Group TitleBest ResponseYou've already chosen the best response.10
Congratz sat! :) You may like to post your solution for others.
 2 years ago

satellite73 Group TitleBest ResponseYou've already chosen the best response.0
not unless everyone is done
 2 years ago

FoolForMath Group TitleBest ResponseYou've already chosen the best response.10
Alright:) Third one is the hardest in my opinion. So well done agian! :)
 2 years ago

anonymoustwo44 Group TitleBest ResponseYou've already chosen the best response.0
uhm for number 2, do you mean the greatest possible value of n for x<0 and y<0?
 2 years ago

bluepig148 Group TitleBest ResponseYou've already chosen the best response.0
Is the answer to the rectangle one 5148?
 2 years ago

bluepig148 Group TitleBest ResponseYou've already chosen the best response.0
@satellite73 Am I right?
 2 years ago

satellite73 Group TitleBest ResponseYou've already chosen the best response.0
that is what i got, yes
 2 years ago

bluepig148 Group TitleBest ResponseYou've already chosen the best response.0
It's been a while, so http://ideone.com/k7o4g My thought process was explained in the comments.
 2 years ago

Ishaan94 Group TitleBest ResponseYou've already chosen the best response.1
For the complimentary problem, \(a^2\)?
 2 years ago

FoolForMath Group TitleBest ResponseYou've already chosen the best response.10
@Ishaan94: \(a^2 \) is not the right answer.
 2 years ago

lgbasallote Group TitleBest ResponseYou've already chosen the best response.0
@FoolForMath do you make these things up? or you have some referneces? coz you would be sooo smart if these are original ___
 2 years ago

perl Group TitleBest ResponseYou've already chosen the best response.0
If the tangent the to the curve x√+y√=a√ at any point on cuts xaxis and yaxis at two distinct points. Can you find the sum of the intercepts ?
 2 years ago

.Sam. Group TitleBest ResponseYou've already chosen the best response.0
answer for (2) is it n=1?
 2 years ago

.Sam. Group TitleBest ResponseYou've already chosen the best response.0
or it could be zero
 2 years ago

FoolForMath Group TitleBest ResponseYou've already chosen the best response.10
#2 answer : 5039 Complimentary problem: \( a \)
 2 years ago

Ishaan94 Group TitleBest ResponseYou've already chosen the best response.1
Oh yeah, a
 2 years ago

experimentX Group TitleBest ResponseYou've already chosen the best response.0
explanation???
 2 years ago
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