Solve: floor(x/5) - floor(x/7) = 1 in Matlab

Sir/Mam, i want to find all the solutions of the equation: floor(x/5) - floor(x/7) = 1 ;where x belongs to Natural Number, so how can i find all its solutions in Matlab? with Thanks

[Zakaria Makhlouki]

 x = 0;
 while x<=100 % how numbers you want verify
    if floor(x/5)-floor(x/7)==1
        x
        break;
    end
    x = x+1;
 end

[Matt J]

You know that there are uncountable solutions, right? For example, all x in the intervals [5,7) and [10,14) satisfy the equation, but [7,10) does not. How would the solution be presented? A list of intervals?

[David Goodmanson]

Hello Sourav,

Your question does not say that the solution has to be a positive integer, so it makes sense to plot it out and see what you have.

x = -50:.001:50;
y = floor(x/5)-floor(x/7);
plot(x,y)

[Geoff Hayes]

@Zakaria - I think that x is a real number and so that would include irrationals as well as the integers (that you have shown above).

[Sourav Kumar]

sorry sir, 'x' would be Natural number in my question, i have also edited my question; please now guide me how to find its solutions as there are now only 17(finite) solutions; with thanks

[Jan]

@Sourav Kumar: Please mention, if this is a homework question. Then posting a solution would not be a true help and answering requires a different strategy.

[Sourav Kumar]

No sir @Jan, it is not a homework question, i just faced it while doing self-math practice; and then wonder whether this type of problem can be solved in Mathlab for self-checking of my answer; that's why i have posted this problem here

[John BG]

Sourav Kumar

this is John BG <mailto:[e-mail] [e-mail]>

Please have a look to my answer and let me know if it solves your question.

[Sourav Kumar]

@John, sorry to inform you sir;your link is directing to a page which says: "Oops! this page doesn't exist";i have also attached an image file of the same

[Jan]

@Sourav: This is not a link. John inserts his name and email address several times in each of his messages. He includes the mail address in < and > characters, such that they are displayed as a link in the forum, which is confusing.

The idea of the forum is to solve problems in public and you do not have to contact contributors personally by email.

[Sourav Kumar]

Thanks a lot sir

Answer 1

[Jan]

Okay, if this is not a homework, than you find enough information here to solve the problem already. I summarize:

• Matt J showed, that the results must be 0 <= x <= 35.
• You can easily check all values:

x      = 0:35;
match  = (floor(x / 5) - floor(x / 7) == 1);
result = x(match)

And here you get your 17 values.

Answer 2

[Matt J]

As I mentioned, the solutions are going to be uncountable because F(x) = floor(x/5) - floor(x/7) is a piecewise constant function of x. However, from a little analysis, we know that they must lie in the interval [0,35]. To see this, note that we have the bounds

      x/5-x/7 +1 >=  floor(x/5) - floor(x/7) >= (x/5-x/7)-1

The upper bound is less than 1 for all x<0 and the lower bound is greater than 1 for all x>35.

Also, F(x) can only change value at multiples of 5 or 7. So, the solution reduces to a search over

   xSet=sort([0:5:35, 0:7:35]);

Once you find the subset of solutions xCritical,

          xCritical = xSet(  floor(xSet/5)-floor(xSet/7) ==1 )

it is a simple matter to look at the full intervals bounded by xSet to see which ones constitute solutions.

[Matt J]

sorry sir, 'x' would be Natural number in my question, i have also edited my question; please now guide me how to find its solutions as there are now only 17(finite) solutions

If the solutions must be naturals, then the problem is even simpler. Just search all the integers from 1 to 35.

[Matt J]

@Jan,

Yes, but I never claimed xCritical was the full set of solutions. Remember, this was when the OP was telling us that the solutions might be non-integer.

Since it looks like a homework problem, I felt we should leave some work for the OP.

[Jan]

@Matt J: I did not see your 2nd last comment, when I typed mine.

I thought about more complicated functions, when a search over all numbers is impossible, and then I've overseen, that this looks like a homework.

I've removed my answer and comment. +1 for yours, with the constraint the "the solution reduces to a search over [xSet]" sounds like this is the full set of results.

[Matt J]

ok, Jan. Thanks for the +1.

Answer 3

[John BG]

Hi Sourav Kumar

this is John BG <mailto:[e-mail] [e-mail]>

x=[1:1:17]
nonzeros(x.*((floor(x/5)-floor(x/7))==1))

   =

       5
       6
      10
      11
      12
      13
      15
      16
      17

if you find this answer useful would you please be so kind to consider marking my answer as Accepted Answer?

To any other reader, if you find this answer useful please consider clicking on the thumbs-up vote link

thanks in advance

John BG

<mailto:[e-mail] [e-mail]>

Answer 4

[Ashen]

plot f(x)=5x+7

[Walter Roberson]

What should we look for in that plot to locate the places where floor(x/5) -f loor(x/7) == 1 ?