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Real Product to Integer (Posted on 2009-09-12) Difficulty: 2 of 5
Determine all possible nonzero real R satisfying R = [R]*{R}, such that 5*{R} - [R]/4 is an integer.

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

See The Solution Submitted by K Sengupta    
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Some Thoughts computer exploration | Comment 1 of 2

When R is positive, [R]*{R} will always be less than R, and never equal. When R is negative, {R} = [R]/([R] - 1).

A table follows for the resulting R (i.e., [R] + {R}), from the values of [R] at decreasing integer values -1, -2, -3, ..., together with 5*{R} - [R]/4:

-.5                           2.75
-1.333333333333333            3.833333333333333
-2.25                         4.5
-3.2                          5
-4.166666666666667            5.416666666666667
-5.142857142857143            5.785714285714286
-6.125                        6.125
-7.111111111111111            6.444444444444445
-8.1                          6.75
-9.090909090909092            7.045454545454545
-10.08333333333333            7.333333333333333
-11.07692307692308            7.615384615384616
-12.07142857142857            7.892857142857143
-13.06666666666667            8.166666666666666
-14.0625                      8.4375
-15.05882352941176            8.705882352941176
-16.05555555555556            8.972222222222221
-17.05263157894737            9.236842105263158
-18.05                        9.5
-19.04761904761905            9.761904761904761
-20.04545454545455            10.02272727272727
-21.04347826086957            10.28260869565217
-22.04166666666667            10.54166666666667
-23.04                        10.8
-24.03846153846154            11.05769230769231
-25.03703703703704            11.31481481481481
-26.03571428571428            11.57142857142857
-27.03448275862069            11.82758620689655
-28.03333333333333            12.08333333333333
-29.03225806451613            12.33870967741936
-30.03125                     12.59375
-31.03030303030303            12.84848484848485
-32.02941176470588            13.10294117647059
-33.02857142857143            13.35714285714286
-34.02777777777778            13.61111111111111
-35.02702702702702            13.86486486486486
-36.02631578947368            14.11842105263158
-37.02564102564103            14.37179487179487
-38.025                       14.625
-39.02439024390244            14.87804878048781
-40.02380952380953            15.13095238095238
-41.02325581395349            15.38372093023256
-42.02272727272727            15.63636363636364
-43.02222222222223            15.88888888888889
-44.02173913043478            16.14130434782609
-45.02127659574468            16.3936170212766
-46.02083333333334            16.64583333333333
-47.02040816326531            16.89795918367347
-48.02                        17.15
-49.01960784313726            17.40196078431373
-50.01923076923077            17.65384615384615
-51.0188679245283             17.90566037735849
-52.01851851851852            18.15740740740741
-53.01818181818182            18.40909090909091
-54.01785714285715            18.66071428571428
-55.01754385964912            18.91228070175439
-56.01724137931034            19.16379310344827
-57.01694915254237            19.41525423728813
-58.01666666666667            19.66666666666667
-59.01639344262295            19.91803278688525
-60.01612903225806            20.16935483870968
-61.01587301587302            20.42063492063492
-62.015625                    20.671875
-63.01538461538462            20.92307692307692
-64.01515151515152            21.17424242424243
-65.01492537313433            21.42537313432836
-66.01470588235294            21.67647058823529
-67.01449275362319            21.92753623188406
-68.01428571428572            22.17857142857143
-69.01408450704226            22.42957746478873
-70.01388888888889            22.68055555555556
-71.01369863013699            22.93150684931507
-72.01351351351352            23.18243243243243
-73.01333333333334            23.43333333333333
-74.01315789473684            23.68421052631579
-75.01298701298701            23.93506493506494
-76.01282051282051            24.18589743589744
-77.01265822784811            24.4367088607595
-78.0125                      24.6875
-79.01234567901234            24.93827160493827
-80.01219512195122            25.1890243902439
-81.01204819277109            25.43975903614458
-82.01190476190476            25.69047619047619
-83.01176470588236            25.94117647058824
-84.01162790697674            26.19186046511628
-85.01149425287356            26.44252873563218
-86.01136363636364            26.69318181818182
-87.01123595505618            26.9438202247191
-88.01111111111111            27.19444444444444
-89.01098901098901            27.44505494505495

It seems that only -3.2 works, producing the integer 5 for 5*{R} - [R]/4.

DEFDBL A-Z
CLS
FOR n = -1 TO -90 STEP -1
 x = n / (n - 1)
 R = n + x
 PRINT R; TAB(30); 5 * x - n / 4
NEXT

using n to represent [R] and x to represent {R}.


  Posted by Charlie on 2009-09-12 15:19:33
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