[Guest]

Determine all possible nonzero real n satisfying n = [n]*{n}, such that 5*{n} - [n]/4 is an integer.

Note: [x] is the greatest integer <= x, and {x} = x -[x]

Edited July 6, 2009 by K Sengupta

[Guest]

Determine all possible nonzero real n satisfying n = [n]*{n}, such that 5*{n} - [n]/4 is an integer.

Note: [x] is the greatest integer <= x, and {x} = x -[x]

there are two: 2.5 and 10.9

[Guest]

as i understand it

it is impossilbe for all positive numbers. for negative numbers the n=[n]{n} is possible once for every whole number (meaning there is one number this happens for in -1 or -2 or ... The only negative number that satisfies the second condition that i see is -3.2

[superprismatic]

Determine all possible nonzero real n satisfying n = [n]*{n}, such that 5*{n} - [n]/4 is an integer.

Note: [x] is the greatest integer <= x, and {x} = x -[x]

-3.2 is the only possibility. It's easy to prove but it is also a little tedious.

[Guest]

Can you be more clear with your question. As I see the way others understood it there seem to be many many solutions:

[n] can be a multiple of 20 and {n} would then be any multiple of 0.2

[n] can be 10 and {n} can be 0.5

[n] can be 6 and {n} can be 0.3 etc

[Guest]

as i understand it

for negative numbers the n=[n]{n} is possible once for every whole number (meaning there is one number this happens for in -1 or -2 or ...

Nice analysis on the whole. But, [n] is the greatest integer <= n, and so n = [n] whenever n is an integer. In that situation if n is a nonzero negative integer then (n) = 0, so that the lhs of n = [n]*{n} is nonzero, but the rhs is zero.

[Guest]

Can you be more clear with your question. As I see the way others understood it there seem to be many many solutions:

[n] can be a multiple of 20 and {n} would then be any multiple of 0.2

[n] can be 10 and {n} can be 0.5

[n] can be 6 and {n} can be 0.3 etc

The problem statement defines, [n] as the greatest integer <= n, and {n} = n -[n]. For example, the greatest integer <= -4.5 is -4, so: [-4.5] = -4 and {-4.5} = 0.5. Similarly, if n = 6.3, then [n]=6, and {n} = 0.3.

Also, if n is any integer, then n = [n], and {n} = 0.

If n= 10.5, then [n] = 10, and {n}= 0.5, and so:[n]*{n} = 5, which is not equal to 10.5, and therefore a contradiction.

The problem statement requires to find all real n that satisfy all the given conditions.

I hope this serves to clarify the fundamental tenets inclusive of the given problem.

[Guest]

For example, the greatest integer <= -4.5 is -4, so: [-4.5] = -4 and {-4.5} = 0.5.

One small correction. This should read:

"For example, the greatest integer <= -4.5 is -5, so: [-4.5] = -5 and {-4.5} = 0.5."

[Guest]

i misscommunicated what i should have said is it happens once in between -1 and -2 once in between -2 and -3 and so on. The first time I was trying to say once with look of -1.****** once with -2.********* and so on.

[Guest]

Determine all possible nonzero real n satisfying n = [n]*{n}, such that 5*{n} - [n]/4 is an integer.

Note: [x] is the greatest integer <= x, and {x} = x -[x]

If I understood the problem right, the only n is -3.2

There are infinitely many values of n that satisfy n = [n] * {n} - these are given by -i*i/(i+1) for any positive non zero integer i. 5*{n}-[n]/4=i*(i+21)/(4*(i+1)). To be an integer, only possibility is i = 4

Edited July 10, 2009 by logician

[Guest]

[Guest]

Good question....

All numbers of the form.....

[x] = -4,-8,-12,.........

and

{x} = [x]/([x] - 1)

[Guest]

Good question....

All numbers of the form.....

[x] = -4,-8,-12,.........

and

{x} = [x]/([x] - 1)

Nice analysis on the whole. But, [n] is the greatest integer <= n, and so n = [n] whenever n is any integer, giving:{n}= n- [n]= 0.

In that situation if n is a nonzero negative integer divisible by 4, then the first condition, that is n = [n]*{n}, is not satisfied, since the lhs is nonzero, but the rhs is zero.

[Guest]

Nice analysis on the whole. But, [n] is the greatest integer <= n, and so n = [n] whenever n is any integer, giving:{n}= n- [n]= 0.

In that situation if n is a nonzero negative integer divisible by 4, then the first condition, that is n = [n]*{n}, is not satisfied, since the lhs is nonzero, but the rhs is zero.

sorry i didn't get what u r saying......is my answer wrong????

[Guest]

sorry i didn't get what u r saying......is my answer wrong????

Not exactly wrong, merely inaccurate. The problem statement requires two conditions to be satisfied, while your solution,that is, "All numbers of the form .[x] = -4,-8,-12,........." merely satisfies the second but fails to comply with the first due to obvious reasons.

However, {x} = [x]/([x] – 1 is a good starting point.

To clarify further, [x] is simply the floor function, and: {x} = x – floor(x).

Edited July 27, 2009 by K Sengupta

[Guest]

Not exactly wrong, merely inaccurate. The problem statement requires two conditions to be satisfied, while your solution,that is, "All numbers of the form .[x] = -4,-8,-12,........." merely satisfies the second but fails to comply with the first due to obvious reasons.

However, {x} = [x]/([x] – 1 is a good starting point.

To clarify further, [x] is simply the floor function, and: {x} = x – floor(x).

[x] = -4

{x} = -4/-5 = 0.8

=> x = -3.2

[x]= -8

{x} = -8/-9 = 8/9

=> x = -64/9

.

.

.

and so on....

so where is x is zero.

[Guest]

[x] = -4

{x} = -4/-5 = 0.8

=> x = -3.2

[x]= -8

{x} = -8/-9 = 8/9

=> x = -64/9

.

.

.

and so on....

Of the various solutions offered, only [x]= -4, {x} = 0.8, giving: x = -3.2 is valid.

A ready check will attest to the fact that the other multiple of 4 values of [x], that is: [x] = -12, -16, ……and so on, with: {x} = [x]/([x] -1) fail to satisfy the second condition.

For example, the solution x = -64/9, ([x]= -8, and {x} = 8/9) is inaccurate, and it fails to satisfy the second condition that:

5*{n} - [n]/4 is an integer, since:

(5*8)/9 + 8/4 = 58/9, which is not an integer.

...so where is x is zero.

I admit that misread the [x] in the answer in terms of post #12 as x, and hence the apparent confusion

.