## 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 non-negative 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 x-axis and y-axis at two distinct points. Can you find the sum of the intercepts ? Good luck! 2 years ago 2 years ago

1. Ishaan94 Group Title

|dw:1333293300537:dw|

2. Ishaan94 Group Title

Why did everyone leave? :(

3. Ishaan94 Group Title

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 $$x\le40$$.

4. FoolForMath Group Title

HINT: Linear Diophantine equation.

5. lgbasallote Group Title

these problems would be unsolveable and fool will greet us April Fool's Day mmhmm <nods> that would be epic

6. FoolForMath Group Title

(1) is very much solvable by almost anybody.

7. experimentX Group Title

both of $$(x, y)$$ are not positive??? either x is negative or y is negative or both are negative?

8. experimentX Group Title

looks like n is greatest common divisor

9. Ishaan94 Group Title

Diophantine sounds like organic chemistry. I'm googling it. 10?

10. FoolForMath Group Title

Lol, he (Diophantus) is the sometimes called the father of algebra.

11. Ishaan94 Group Title

It took me time but the Linear Diophantine thing is really nice. Thanks.

12. experimentX Group Title

is the answer of 2 1??

13. FoolForMath Group Title

1 is the not the right answer.

14. AravindG Group Title

......

15. experimentX Group Title

must be infinity then

16. FoolForMath Group Title

No it's not infinity.

17. 2bornot2b Group Title

There are 11 non negative integral solutions to the given diophantine equation

18. 2bornot2b Group Title

Am I right?

19. FoolForMath Group Title

Well done @2bornot2b!

20. 2bornot2b Group Title

Thanks!

21. FoolForMath Group Title

If other want, you may like to the post the solution too.

22. 2bornot2b Group Title

Using euclidean reduction it can be shown that the integral solutions are of the form $$x=4n$$ and $$y=-3(n-10)$$ 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 (n-10) must be negative so as to have positive y, which is possible for n=0,1,2,3....,10. So we have 11 solutions

23. 2bornot2b Group Title

And yes for n-10=0, the value is true too.

24. Ishaan94 Group Title

For 2nd 81*64?

25. FoolForMath Group Title

@Ishaan94: Ishaan you are very close.

26. Ishaan94 Group Title

81*64 - 1? lol

27. FoolForMath Group Title

haha, nopes :)

28. Ishaan94 Group Title

0?

29. experimentX Group Title

i was thinking the same .. lol

30. experimentX Group Title

but it seems 0 is nowhere close to 81*64

31. Ishaan94 Group Title

Yeah lol

32. Mr.Math Group Title

Isn't it obvious that if both $$x$$ and $$y$$ are non-positive, then the largest possible value of $$n$$ is $$0$$?

33. Mr.Math Group Title

I'm talking about the second problem.

34. experimentX Group Title

yeah ... but both $$(x, y)$$ are not positive. might mean both cannot be ++, but can be +- or -+ ... just considering possibility

35. Ishaan94 Group Title

It is, but at first I presumed it was both x and y can't be non-positive integral :/

36. Ishaan94 Group Title

If you'll read the first question you might understand the cause of my presumption :/

37. experimentX Group Title

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

38. Mr.Math Group Title

Oh he probably meant "such that x and y are not both positive", meaning that only one of them can be positive.

39. experimentX Group Title

in that case .. Mr. Math must be right.

40. Ishaan94 Group Title

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|

41. Mr.Math Group Title

The answer of (2) is $$0$$, unless something's wrong with the question.

42. Ishaan94 Group Title

FoolForMath for the 3rd question, Can you tell the value of r(3,2)?

43. experimentX Group Title

@FoolForMath what do you say??

44. Ishaan94 Group Title

FoolForMath WHAT SAY YOU? *This is more cinematic hehe*

45. Mr.Math Group Title

What does "is tiled with" mean? (My English is failing me)!

46. Ishaan94 Group Title

47. Mr.Math Group Title

Oh I see.

48. Mr.Math Group Title

But why is $$r(2,1)=3$$?

49. FoolForMath Group Title

Ishaan, $$r(3,2)=18$$.

50. satellite73 Group Title

no you are trying to find n is r(3,2)=16?

51. Mr.Math Group Title

@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 non-positive, then we can choose $$x>>> 0\ge y$$ and then n would have no maximum.

52. experimentX Group Title

number of rectangles?? you mean there are lot's of rectangle???

53. satellite73 Group Title

not both postive ok i see r(3,2)=18

54. FoolForMath Group Title

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, 81-64 = 17 you have to maximize this value.

55. satellite73 Group Title

?? lost me on that one

56. experimentX Group Title

x=inf, y=-1, n = inf

57. satellite73 Group Title

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

58. Ishaan94 Group Title

But as long as you have positive y intercept and x intercept, you will always have positive solutions for (x,y) subject to $$x,y \in \mathbb{R}$$.

59. FoolForMath Group Title

sat, if x and y are non-negative integers, then the the greatest integer that cannot be written in the form ax+by is ab − a − b. [assuming (a,b)=1 ]

60. experimentX Group Title

61. satellite73 Group Title

i have an answer i like for the second to last one

62. satellite73 Group Title

whew. left the house, had a beer and thought about it a little more clearly

63. FoolForMath Group Title

Congratz sat! :) You may like to post your solution for others.

64. satellite73 Group Title

not unless everyone is done

65. FoolForMath Group Title

Alright:) Third one is the hardest in my opinion. So well done agian! :)

66. satellite73 Group Title

merci

67. anonymoustwo44 Group Title

uhm for number 2, do you mean the greatest possible value of n for x<0 and y<0?

68. bluepig148 Group Title

Is the answer to the rectangle one 5148?

69. bluepig148 Group Title

@satellite73 Am I right?

70. satellite73 Group Title

that is what i got, yes

71. bluepig148 Group Title

It's been a while, so http://ideone.com/k7o4g My thought process was explained in the comments.

72. Ishaan94 Group Title

For the complimentary problem, $$a^2$$?

73. FoolForMath Group Title

@Ishaan94: $$a^2$$ is not the right answer.

74. lgbasallote Group Title

@FoolForMath do you make these things up? or you have some referneces? coz you would be sooo smart if these are original -___-

75. perl Group Title

If the tangent the to the curve x√+y√=a√ at any point on cuts x-axis and y-axis at two distinct points. Can you find the sum of the intercepts ?

76. .Sam. Group Title

hmm

77. .Sam. Group Title

answer for (2) is it n=-1?

78. .Sam. Group Title

or it could be zero

79. FoolForMath Group Title

#2 answer : 5039 Complimentary problem: $$a$$

80. Ishaan94 Group Title

Oh yeah, a

81. experimentX Group Title

explanation???