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 Real Product to Integer (Posted on 2009-09-12)
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 No Rating

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 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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