How many positive integers N less than 1000 are there such that it can be expressed as x ⌊ x ⌋ for some real x ?
Notation:
⌊
⋅
⌋
denotes the
floor function
.
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For [x]=2 how you will get any N between 4 and 9 ???
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2 ⌊ x ⌋ = 2 2 anything between 4 and 9.
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Suppose x = 2 . 5 so ( 2 . 5 ) [ 2 . 5 ] = 2 . 5 2 which is not an integer
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@Kushal Bose – But x = 2 . 5 is not an integer either. The question asks for how many positive integers.
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@Hana Wehbi – Then how all integers between 4 to 9 can be achieved ?
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@Kushal Bose – 4 , 5 , 6 , 7 , 8 aren't these are 5 positive integers between 2 2 and 3 2 excluding 3 2 .
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First x must be less than 5 , since otherwise x ⌊ x ⌋ would be 3 1 2 5 which is greater than 1 0 0 0 .
Because ⌊ x ⌋ must be an integer, we have to do the following:
For ⌊ x ⌋ = 0 , N = 1 , as long as x = 0 . This gives us 1 value of N .
For ⌊ x ⌋ = 1 , N can be anything between 1 1 and 2 1 excluding 2 1 .
Therefore, N = 1 . However, we got N = 1 in case 1 so it got counted twice.
For ⌊ x ⌋ = 2 , N can be anything between 2 2 to 3 2 excluding 3 2 . This gives us 3 2 − 2 2 = 5 N ′ s
For ⌊ x ⌋ = 3 , N can be anything between 3 3 to 4 3 excluding 4 3 . This gives us 4 3 − 3 3 = 3 7 N ′ s .
For ⌊ x ⌋ = 4 , N can be anything between 4 4 to 5 4 excluding 5 4 . This gives us
5 4 − 4 4 = 3 6 9 N ′ s . We stop here since x < 5 . Thus the possible answers for N = 1 + 5 + 3 7 + 3 6 9 = 4 1 2 .