Pandigital numbers | ArithmeticMean[],GeometricMean[]

 
 
Each of the positive integers   A, \; B, \; C, \; D   is pandigital such that

>>   A \; < \; B \; < \; C \; < \; D
>>   A \; + \; D   is also pandigital
>>   B \; = \; GeometricMean \,[A, \; D]
>>   C \; = \; ArithmeticMean \,[A, \; D]

 
Find   A, \; B, \; C, \; D

 

Here's one solution:
 

A \; = \; 1076539482

B \; = \; 2153078964 \; = \; GeometricMean \,[ \,1076539482, \; 4306157928 \,]

C \; = \; 2691348705 \; = \; ArithmeticMean \,[1076539482, \; 4306157928 \,]

D \; = \; 4306157928

A \; + \; D \; = \; 1076539482 \; + \; 4306157928 \; = \; 5382697410

Each of the integers   A, \; B, \; C, \; D   is pandigital

 

Find other solutions.
 

 
 
 
Paul found:

 

where   A + D   is ten times a pandigital number:

\{2107653948,4215307896,5269134870,8430615792,10538269740 \}
\{2136507948,4273015896,5341269870,8546031792,10682539740 \}
\{2136759048,4273518096,5341897620,8547036192,10683795240 \}
\{2147936508,4295873016,5369841270,8591746032,10739682540 \}
\{2150793648,4301587296,5376984120,8603174592,10753968240 \}
\{2197530864,4395061728,5493827160,8790123456,10987654320 \}
\{2409513678,4819027356,6023784195,9638054712,12047568390 \}

 
And where   A + D   is a pandigital number:

\{1076539482,2153078964,2691348705,4306157928,5382697410 \}
\{1098765432,2197530864,2746913580,4395061728,5493827160 \}
\{1234567890,2469135780,3086419725,4938271560,6172839450 \}
\{1234568790,2469137580,3086421975,4938275160,6172843950 \}
\{1269835704,2539671408,3174589260,5079342816,6349178520 \}
\{1270356984,2540713968,3175892460,5081427936,6351784920 \}
\{1274590368,2549180736,3186475920,5098361472,6372951840 \}
\{1298435670,2596871340,3246089175,5193742680,6492178350 \}
\{1365079482,2730158964,3412698705,5460317928,6825397410 \}
\{1367590482,2735180964,3418976205,5470361928,6837952410 \}
\{1472590368,2945180736,3681475920,5890361472,7362951840 \}
\{1476859032,2953718064,3692147580,5907436128,7384295160 \}
\{1479365082,2958730164,3698412705,5917460328,7396825410 \}
\{1564329870,3128659740,3910824675,6257319480,7821649350 \}
\{1587296430,3174592860,3968241075,6349185720,7936482150 \}
\{1274590368,3186475920,3186475920,5098361472,6372951840 \}
\{1507936482,3015872964,3769841205,6031745928,7539682410 \}
\{1526437098,3052874196,3816092745,6105748392,7632185490 \}
\{1642958730,3285917460,4107396825,6571834920,8214793650 \}
\{1705483926,3410967852,4263709815,6821935704,8527419630 \}
\{1728439506,3456879012,4321098765,6913758024,8642197530 \}
\{1759048632,3518097264,4397621580,7036194528,8795243160 \}
\{1472590368,3681475920,3681475920,5890361472,7362951840 \}
\{1847356290,3694712580,4618390725,7389425160,9236781450 \}
\{1950637284,3901274568,4876593210,7802549136,9753186420 \}
\{1962530874,3925061748,4906327185,7850123496,9812654370 \}
\{1962730854,3925461708,4906827135,7850923416,9813654270 \}
\{1962850734,3925701468,4907126835,7851402936,9814253670 \}
\{1975308642,3950617284,4938271605,7901234568,9876543210 \}

 
 
 

A \; + \; D \; = \; B
ArithmeticMean \, [A, \; D] \; = \; C
GeometricMean \, [A, \; D] \; = \; E

Format   \{A, \; B, \; C, \; D, \; E\}

\{1098765432,5493827160,2746913580,4395061728,2197530864\}
\{1269835704,6349178520,3174589260,5079342816,2539671408\}
\{1270356984,6351784920,3175892460,5081427936,2540713968\}
\{1274590368,6372951840,3186475920,5098361472,2549180736\}
\{1472590368,7362951840,3681475920,5890361472,2945180736\}
\{1476859032,7384295160,3692147580,5907436128,2953718064\}
\{1098765432,7691358024,2746913580,4395061728,2197530864\}
\{1098765432,8790123456,2746913580,4395061728,2197530864\}
\{1759048632,8795243160,4397621580,7036194528,3518097264\}
\{1950637284,9753186420,4876593210,7802549136,3901274568\}
\{1234567890,9876543120,3086419725,4938271560,2469135780\}

 
 
 
 
 
 
 
 
 
 
 
 
 
 

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About benvitalis

math grad - Interest: Number theory
This entry was posted in Number Puzzles and tagged , , . Bookmark the permalink.

5 Responses to Pandigital numbers | ArithmeticMean[],GeometricMean[]

  1. paul says:

    Here are all of them ( can’t guarantee that though 🙂 ).

    Format is {a, b, c, d}.

    {1076539482,2153078964,2691348705,4306157928}
    {1098765432,2197530864,2746913580,4395061728}
    {1234567890,2469135780,3086419725,4938271560}
    {1234568790,2469137580,3086421975,4938275160}
    {1269835704,2539671408,3174589260,5079342816}
    {1270356984,2540713968,3175892460,5081427936}
    {1274590368,2549180736,3186475920,5098361472}
    {1298435670,2596871340,3246089175,5193742680}
    {1365079482,2730158964,3412698705,5460317928}
    {1367590482,2735180964,3418976205,5470361928}
    {1472590368,2945180736,3681475920,5890361472}
    {1476859032,2953718064,3692147580,5907436128}
    {1479365082,2958730164,3698412705,5917460328}
    {1564329870,3128659740,3910824675,6257319480}
    {1587296430,3174592860,3968241075,6349185720}
    {1274590368,3186475920,3186475920,5098361472}
    {1507936482,3015872964,3769841205,6031745928}
    {1526437098,3052874196,3816092745,6105748392}
    {1642958730,3285917460,4107396825,6571834920}
    {1705483926,3410967852,4263709815,6821935704}
    {1728439506,3456879012,4321098765,6913758024}
    {1759048632,3518097264,4397621580,7036194528}
    {1472590368,3681475920,3681475920,5890361472}
    {1847356290,3694712580,4618390725,7389425160}
    {1950637284,3901274568,4876593210,7802549136}
    {1962530874,3925061748,4906327185,7850123496}
    {1962730854,3925461708,4906827135,7850923416}
    {1962850734,3925701468,4907126835,7851402936}
    {1975308642,3950617284,4938271605,7901234568}
    {2107653948,4215307896,5269134870,8430615792}
    {2136507948,4273015896,5341269870,8546031792}
    {2136759048,4273518096,5341897620,8547036192}
    {2147936508,4295873016,5369841270,8591746032}
    {2150793648,4301587296,5376984120,8603174592}
    {2197530864,4395061728,5493827160,8790123456}
    {2409513678,4819027356,6023784195,9638054712}

    Paul.

    • benvitalis says:

      I had 29 solutions in mind, such that A + D is also pandigital.
      You found 36. The seven extra solutions are also interesting since each of them is
      10 times a pandigital number.

      • benvitalis says:

        Your last seven solutions being:

        {2107653948,4215307896,5269134870,8430615792}
        2107653948+8430615792 = 10538269740

        {2136507948,4273015896,5341269870,8546031792}
        2136507948+8546031792 = 10682539740

        {2136759048,4273518096,5341897620,8547036192}
        2136759048+8547036192 = 10683795240

        {2147936508,4295873016,5369841270,8591746032}
        2147936508+8591746032 = 10739682540

        {2150793648,4301587296,5376984120,8603174592}
        2150793648+8603174592 = 10753968240

        {2197530864,4395061728,5493827160,8790123456}
        2197530864+8790123456 = 10987654320

        {2409513678,4819027356,6023784195,9638054712}
        2409513678+9638054712 = 12047568390

  2. paul says:

    Here are a few more that aren’t with a<b<c<d.

    A + D = B and the arithmetic mean of A & D is = C

    1098765432 5493827160 2746913580 4395061728
    1269835704 6349178520 3174589260 5079342816
    1270356984 6351784920 3175892460 5081427936
    1274590368 6372951840 3186475920 5098361472
    1472590368 7362951840 3681475920 5890361472
    1476859032 7384295160 3692147580 5907436128
    1098765432 7691358024 2746913580 4395061728
    1098765432 8790123456 2746913580 4395061728
    1759048632 8795243160 4397621580 7036194528
    1950637284 9753186420 4876593210 7802549136
    1234567890 9876543120 3086419725 4938271560
    the geometric mean od A & D is

    2197530864
    2539671408
    2540713968
    2549180736
    2945180736
    2953718064
    2197530864
    2197530864
    3518097264
    3901274568
    2469135780

    Paul.

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