Question

(1) Proove that the equation .

[x] + [2x] + [4x] + [8x] + [16x] + [32] = 12345 has no solution.


(2) Find the no. of non negative Solution , [x/2]+ [x/3]+[x/5]= x, x>= 0
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Answer 1

Let x = i + f where i is the integer part or integral part and f the fractional part of x. We have f < 1, and

[x] + [2x] + [4x] + [8x] + [16x] + [3

2x] = 63i + [f] + [2f] + [4f] + [8f] + [16f] + [32f]

since f < 1, [f] = 0, and we now have

63i + [f] + [2f] + [4f] + [8f] + [16f] + [32f] = 63i + [2f] + [4f] + [8f] + [16f] + [32f] = 12345 = 63 × 195 + 60

So i = 195, and [2f] + [4f] + [8f] + [16f] + [32f] = 60 (*)

Since max[nf] = n - 1, the maximum value of [2f] + [4f] + [8f] + [16f] + [32f] = 1 + 3 + 7 + 15 + 31 = 57. Therefore, equation (*) is not possible, and there is no f that can satisfy the equation in the problem, and thus there is no x.

Answer 2

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Let x = i + f where i is the integer part or integral part and f the fractional part of x. We have f < 1, and

[x] + [2x] + [4x] + [8x] + [16x] + [3

2x] = 63i + [f] + [2f] + [4f] + [8f] + [16f] + [32f]

since f < 1, [f] = 0, and we now have

63i + [f] + [2f] + [4f] + [8f] + [16f] + [32f] = 63i + [2f] + [4f] + [8f] + [16f] + [32f] = 12345 = 63 × 195 + 60

So i = 195, and [2f] + [4f] + [8f] + [16f] + [32f] = 60 (*)

Since max[nf] = n - 1, the maximum value of [2f] + [4f] + [8f] + [16f] + [32f] = 1 + 3 + 7 + 15 + 31 = 57. Therefore, equation (*) is not possible, and there is no f that can satisfy the equation in the problem, and thus there is no x.

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