Quantcast

A community for students. Sign up today!

Here's the question you clicked on:

55 members online
  • 0 replying
  • 0 viewing

FoolForMath

  • 2 years ago

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!

  • This Question is Closed
  1. Ishaan94
    • 2 years ago
    Best Response
    You've already chosen the best response.
    Medals 1

    |dw:1333293300537:dw|

  2. Ishaan94
    • 2 years ago
    Best Response
    You've already chosen the best response.
    Medals 1

    Why did everyone leave? :(

  3. Ishaan94
    • 2 years ago
    Best Response
    You've already chosen the best response.
    Medals 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 \(x\le40\).

  4. FoolForMath
    • 2 years ago
    Best Response
    You've already chosen the best response.
    Medals 10

    HINT: Linear Diophantine equation.

  5. lgbasallote
    • 2 years ago
    Best Response
    You've already chosen the best response.
    Medals 0

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

  6. FoolForMath
    • 2 years ago
    Best Response
    You've already chosen the best response.
    Medals 10

    (1) is very much solvable by almost anybody.

  7. experimentX
    • 2 years ago
    Best Response
    You've already chosen the best response.
    Medals 0

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

  8. experimentX
    • 2 years ago
    Best Response
    You've already chosen the best response.
    Medals 0

    looks like n is greatest common divisor

  9. Ishaan94
    • 2 years ago
    Best Response
    You've already chosen the best response.
    Medals 1

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

  10. FoolForMath
    • 2 years ago
    Best Response
    You've already chosen the best response.
    Medals 10

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

  11. Ishaan94
    • 2 years ago
    Best Response
    You've already chosen the best response.
    Medals 1

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

  12. experimentX
    • 2 years ago
    Best Response
    You've already chosen the best response.
    Medals 0

    is the answer of 2 1??

  13. FoolForMath
    • 2 years ago
    Best Response
    You've already chosen the best response.
    Medals 10

    1 is the not the right answer.

  14. AravindG
    • 2 years ago
    Best Response
    You've already chosen the best response.
    Medals 0

    ......

  15. experimentX
    • 2 years ago
    Best Response
    You've already chosen the best response.
    Medals 0

    must be infinity then

  16. FoolForMath
    • 2 years ago
    Best Response
    You've already chosen the best response.
    Medals 10

    No it's not infinity.

  17. 2bornot2b
    • 2 years ago
    Best Response
    You've already chosen the best response.
    Medals 2

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

  18. 2bornot2b
    • 2 years ago
    Best Response
    You've already chosen the best response.
    Medals 2

    Am I right?

  19. FoolForMath
    • 2 years ago
    Best Response
    You've already chosen the best response.
    Medals 10

    Well done @2bornot2b!

  20. 2bornot2b
    • 2 years ago
    Best Response
    You've already chosen the best response.
    Medals 2

    Thanks!

  21. FoolForMath
    • 2 years ago
    Best Response
    You've already chosen the best response.
    Medals 10

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

  22. 2bornot2b
    • 2 years ago
    Best Response
    You've already chosen the best response.
    Medals 2

    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
    • 2 years ago
    Best Response
    You've already chosen the best response.
    Medals 2

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

  24. Ishaan94
    • 2 years ago
    Best Response
    You've already chosen the best response.
    Medals 1

    For 2nd 81*64?

  25. FoolForMath
    • 2 years ago
    Best Response
    You've already chosen the best response.
    Medals 10

    @Ishaan94: Ishaan you are very close.

  26. Ishaan94
    • 2 years ago
    Best Response
    You've already chosen the best response.
    Medals 1

    81*64 - 1? lol

  27. FoolForMath
    • 2 years ago
    Best Response
    You've already chosen the best response.
    Medals 10

    haha, nopes :)

  28. Ishaan94
    • 2 years ago
    Best Response
    You've already chosen the best response.
    Medals 1

    0?

  29. experimentX
    • 2 years ago
    Best Response
    You've already chosen the best response.
    Medals 0

    i was thinking the same .. lol

  30. experimentX
    • 2 years ago
    Best Response
    You've already chosen the best response.
    Medals 0

    but it seems 0 is nowhere close to 81*64

  31. Ishaan94
    • 2 years ago
    Best Response
    You've already chosen the best response.
    Medals 1

    Yeah lol

  32. Mr.Math
    • 2 years ago
    Best Response
    You've already chosen the best response.
    Medals 0

    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
    • 2 years ago
    Best Response
    You've already chosen the best response.
    Medals 0

    I'm talking about the second problem.

  34. experimentX
    • 2 years ago
    Best Response
    You've already chosen the best response.
    Medals 0

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

  35. Ishaan94
    • 2 years ago
    Best Response
    You've already chosen the best response.
    Medals 1

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

  36. Ishaan94
    • 2 years ago
    Best Response
    You've already chosen the best response.
    Medals 1

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

  37. experimentX
    • 2 years ago
    Best Response
    You've already chosen the best response.
    Medals 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

  38. Mr.Math
    • 2 years ago
    Best Response
    You've already chosen the best response.
    Medals 0

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

  39. experimentX
    • 2 years ago
    Best Response
    You've already chosen the best response.
    Medals 0

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

  40. Ishaan94
    • 2 years ago
    Best Response
    You've already chosen the best response.
    Medals 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|

  41. Mr.Math
    • 2 years ago
    Best Response
    You've already chosen the best response.
    Medals 0

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

  42. Ishaan94
    • 2 years ago
    Best Response
    You've already chosen the best response.
    Medals 1

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

  43. experimentX
    • 2 years ago
    Best Response
    You've already chosen the best response.
    Medals 0

    @FoolForMath what do you say??

  44. Ishaan94
    • 2 years ago
    Best Response
    You've already chosen the best response.
    Medals 1

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

  45. Mr.Math
    • 2 years ago
    Best Response
    You've already chosen the best response.
    Medals 0

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

  46. Ishaan94
    • 2 years ago
    Best Response
    You've already chosen the best response.
    Medals 1

    Hmm made up of, maybe

  47. Mr.Math
    • 2 years ago
    Best Response
    You've already chosen the best response.
    Medals 0

    Oh I see.

  48. Mr.Math
    • 2 years ago
    Best Response
    You've already chosen the best response.
    Medals 0

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

  49. FoolForMath
    • 2 years ago
    Best Response
    You've already chosen the best response.
    Medals 10

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

  50. satellite73
    • 2 years ago
    Best Response
    You've already chosen the best response.
    Medals 0

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

  51. Mr.Math
    • 2 years ago
    Best Response
    You've already chosen the best response.
    Medals 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 non-positive, then we can choose \(x>>> 0\ge y\) and then n would have no maximum.

  52. experimentX
    • 2 years ago
    Best Response
    You've already chosen the best response.
    Medals 0

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

  53. satellite73
    • 2 years ago
    Best Response
    You've already chosen the best response.
    Medals 0

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

  54. FoolForMath
    • 2 years ago
    Best Response
    You've already chosen the best response.
    Medals 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, 81-64 = 17 you have to maximize this value.

  55. satellite73
    • 2 years ago
    Best Response
    You've already chosen the best response.
    Medals 0

    ?? lost me on that one

  56. experimentX
    • 2 years ago
    Best Response
    You've already chosen the best response.
    Medals 0

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

  57. satellite73
    • 2 years ago
    Best Response
    You've already chosen the best response.
    Medals 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

  58. Ishaan94
    • 2 years ago
    Best Response
    You've already chosen the best response.
    Medals 1

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

  59. FoolForMath
    • 2 years ago
    Best Response
    You've already chosen the best response.
    Medals 10

    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
    • 2 years ago
    Best Response
    You've already chosen the best response.
    Medals 0

    x=inf, y=-1, n = inf what about this ??

  61. satellite73
    • 2 years ago
    Best Response
    You've already chosen the best response.
    Medals 0

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

  62. satellite73
    • 2 years ago
    Best Response
    You've already chosen the best response.
    Medals 0

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

  63. FoolForMath
    • 2 years ago
    Best Response
    You've already chosen the best response.
    Medals 10

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

  64. satellite73
    • 2 years ago
    Best Response
    You've already chosen the best response.
    Medals 0

    not unless everyone is done

  65. FoolForMath
    • 2 years ago
    Best Response
    You've already chosen the best response.
    Medals 10

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

  66. satellite73
    • 2 years ago
    Best Response
    You've already chosen the best response.
    Medals 0

    merci

  67. anonymoustwo44
    • 2 years ago
    Best Response
    You've already chosen the best response.
    Medals 0

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

  68. bluepig148
    • 2 years ago
    Best Response
    You've already chosen the best response.
    Medals 0

    Is the answer to the rectangle one 5148?

  69. bluepig148
    • 2 years ago
    Best Response
    You've already chosen the best response.
    Medals 0

    @satellite73 Am I right?

  70. satellite73
    • 2 years ago
    Best Response
    You've already chosen the best response.
    Medals 0

    that is what i got, yes

  71. bluepig148
    • 2 years ago
    Best Response
    You've already chosen the best response.
    Medals 0

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

  72. Ishaan94
    • 2 years ago
    Best Response
    You've already chosen the best response.
    Medals 1

    For the complimentary problem, \(a^2\)?

  73. FoolForMath
    • 2 years ago
    Best Response
    You've already chosen the best response.
    Medals 10

    @Ishaan94: \(a^2 \) is not the right answer.

  74. lgbasallote
    • 2 years ago
    Best Response
    You've already chosen the best response.
    Medals 0

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

  75. perl
    • 2 years ago
    Best Response
    You've already chosen the best response.
    Medals 0

    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.
    • 2 years ago
    Best Response
    You've already chosen the best response.
    Medals 0

    hmm

  77. .Sam.
    • 2 years ago
    Best Response
    You've already chosen the best response.
    Medals 0

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

  78. .Sam.
    • 2 years ago
    Best Response
    You've already chosen the best response.
    Medals 0

    or it could be zero

  79. FoolForMath
    • 2 years ago
    Best Response
    You've already chosen the best response.
    Medals 10

    #2 answer : 5039 Complimentary problem: \( a \)

  80. Ishaan94
    • 2 years ago
    Best Response
    You've already chosen the best response.
    Medals 1

    Oh yeah, a

  81. experimentX
    • 2 years ago
    Best Response
    You've already chosen the best response.
    Medals 0

    explanation???

  82. Not the answer you are looking for?
    Search for more explanations.

    Search OpenStudy
    • Attachments:

Ask your own question

Ask a Question
Find more explanations on OpenStudy

Your question is ready. Sign up for free to start getting answers.

spraguer (Moderator)
5 → View Detailed Profile

is replying to Can someone tell me what button the professor is hitting...

23

  • Teamwork 19 Teammate
  • Problem Solving 19 Hero
  • You have blocked this person.
  • ✔ You're a fan Checking fan status...

Thanks for being so helpful in mathematics. If you are getting quality help, make sure you spread the word about OpenStudy.

This is the testimonial you wrote.
You haven't written a testimonial for Owlfred.