Guest Posted July 6, 2009 Report Share Posted July 6, 2009 (edited) 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 Quote Link to comment Share on other sites More sharing options...
0 Guest Posted July 6, 2009 Report Share Posted July 6, 2009 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 Quote Link to comment Share on other sites More sharing options...
0 Guest Posted July 6, 2009 Report Share Posted July 6, 2009 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 Quote Link to comment Share on other sites More sharing options...
0 superprismatic Posted July 6, 2009 Report Share Posted July 6, 2009 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. Quote Link to comment Share on other sites More sharing options...
0 Guest Posted July 6, 2009 Report Share Posted July 6, 2009 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 Quote Link to comment Share on other sites More sharing options...
0 Guest Posted July 7, 2009 Report Share Posted July 7, 2009 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. Quote Link to comment Share on other sites More sharing options...
0 Guest Posted July 7, 2009 Report Share Posted July 7, 2009 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. Quote Link to comment Share on other sites More sharing options...
0 Guest Posted July 7, 2009 Report Share Posted July 7, 2009 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." Quote Link to comment Share on other sites More sharing options...
0 Guest Posted July 7, 2009 Report Share Posted July 7, 2009 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. Quote Link to comment Share on other sites More sharing options...
0 Guest Posted July 10, 2009 Report Share Posted July 10, 2009 (edited) 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 Quote Link to comment Share on other sites More sharing options...
0 Guest Posted July 11, 2009 Report Share Posted July 11, 2009 Quote Link to comment Share on other sites More sharing options...
0 Guest Posted July 25, 2009 Report Share Posted July 25, 2009 Good question.... All numbers of the form..... [x] = -4,-8,-12,......... and {x} = [x]/([x] - 1) Quote Link to comment Share on other sites More sharing options...
0 Guest Posted July 26, 2009 Report Share Posted July 26, 2009 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. Quote Link to comment Share on other sites More sharing options...
0 Guest Posted July 27, 2009 Report Share Posted July 27, 2009 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???? Quote Link to comment Share on other sites More sharing options...
0 Guest Posted July 27, 2009 Report Share Posted July 27, 2009 (edited) 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 Quote Link to comment Share on other sites More sharing options...
0 Guest Posted July 28, 2009 Report Share Posted July 28, 2009 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. Quote Link to comment Share on other sites More sharing options...
0 Guest Posted July 28, 2009 Report Share Posted July 28, 2009 [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. Quote Link to comment Share on other sites More sharing options...
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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 by K SenguptaLink to comment
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