Exploring the Non-Integral Solutions of the Greatest Integer Function

In summary, the equation [x][y] = x + y has integer solutions of (0, 0) and (2, 2). To find the non-integer solutions, we can rewrite the equation as xy = x + y and solve for y. This gives us the following possibilities: y = x/(x-1) if x is not 1, y = 1 + y if x = 1, and y = 0 if x = 0. From these, we can see that the only other possible solution is (0, 1) but this does not satisfy the original equation. Therefore, the only integer solutions are (0, 0) and (2, 2).
  • #1
sadhu
157
0
can anyone tell me how to solve for integer solutions of

[x]*[y]=x+y

tell the interval of its non integral solutions

pleazzzzzzzzzzzzzzzzzz...
 
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  • #2
But [x] = x for all integers x, so the integer solutions of [x][y] = x + y are the same as the integer solutions of xy = x + y, (x, y) in {(0, 0), (2, 2)}.
 
  • #3
well can you show the step

the answer is (0,0) ,(2,2)

i am looking for good and proper stepwise answer and not guesses , i think my method is very weak ..thats why i am here
 
  • #4
first, I must say that "Guessing" (and then checking your guess) is a perfectly "good and proper" method! For n an integer, [n]= n so your equation is simply xy= x+ y. You can write this as xy- y= (x-1)y= x or, if x is not 1, y= x/(x-1). If x is not 0, that says x-1 divides x. The only integer x such that x-1 is a factor of x, is x= 2. You can then check x= 1 or x= 0 separately: If x= 1, xy= x+ y becomes y= 1+ y which is never true.; If x= 0, then xy= x+ y becomes 0= 0+ y which is true for y= 0. The only only solutions are x= y= 0 and x= y= 2.
 
  • #5
i agree to what you said

if you replace x-1=k
y=1+1/k
case 1
k>=1

y=2
no further value of k will do as 1/k is a fraction and goes on to decrease

similiarly you can do k<1

k=-1
y=0

rest all give fraction of decreasing value

ok this much is clear to me but i can't even think of something to start with for next part
 

1. What is the "Greatest Integer function" or "floor function"?

The Greatest Integer function, denoted by ⌈ x ⌉ or [x], rounds down any real number x to the nearest integer that is less than or equal to x. In other words, it gives the greatest integer that is less than or equal to x.

2. How is the "Greatest Integer function" different from rounding down?

Rounding down simply drops any digits after the decimal point, while the Greatest Integer function rounds down to the nearest integer. For example, ⌈3.8⌉ = 3, while rounding down 3.8 gives 3.

3. What are some practical applications of the "Greatest Integer function"?

The Greatest Integer function is commonly used in computer programming to convert a real number to an integer. It is also used in mathematics to define piecewise functions, to solve inequalities, and to express the floor function in terms of other mathematical functions.

4. What is the domain and range of the "Greatest Integer function"?

The domain of the Greatest Integer function is all real numbers, while the range consists of only integers. This means that any real number can be input into the function, but the output will always be an integer.

5. Is there any relation between the "Greatest Integer function" and the "Ceiling function"?

Yes, the Greatest Integer function and the Ceiling function are complementary to each other. While the Greatest Integer function rounds down to the nearest integer, the Ceiling function rounds up to the nearest integer. In other words, ⌈ x ⌉ + ⌈ x ⌉ = x + 1.

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