Re: Numbers and their reversals

• To: mathgroup at smc.vnet.net
• Subject: [mg53715] Re: [mg53687] Numbers and their reversals
• From: DrBob <drbob at bigfoot.com>
• Date: Mon, 24 Jan 2005 03:37:47 -0500 (EST)
• References: <200501230702.CAA11076@smc.vnet.net> <opsk21ldpjiz9bcq@monster.ma.dl.cox.net>
• Sender: owner-wri-mathgroup at wolfram.com

```By the way, 91 isn't the most frequent factor in these problems.

Here are the top ten factors for products up to six digits (with counts):

Timing@Take[Sort@Frequencies@Flatten[omariD /@ Range@999999], -10]

{46.797*Second, {{286, 182},
{605, 22}, {676, 4},
{891, 1001}, {1151, 111},
{1951, 91}, {2921, 101},
{3969, 3}, {15376, 2},
{16593, 11}}}

91 appears in 1951 cases.

Bobby

On Sun, 23 Jan 2005 19:11:27 -0600, DrBob <drbob at bigfoot.com> wrote:

> (Some post as before, but there may have been a bad character in place of "!=".)
>
> There are actually LOTS of solutions.
>
> Here's a verification of the ones you listed, including OTHER factors that work (other than 17 and 91). I'm not counting trivial cases where the product is its own reverse.
>
> Clear[omariQ, omariD]
> reverse = FromDigits@Reverse@IntegerDigits@# &;
> commonDivisors = Rest@Divisors@GCD[#, reverse@#] &;
> omariQ[n_Integer,
>    k_Integer] := n != reverse@n && IntegerQ[n/k] && reverse[n/k] == reverse[n]/k
> omariD[n_Integer] := Select[commonDivisors@n, omariQ[n, #] &]
> omariQ[n_Integer] := Length@omariD@n > 0
> omariD /@ {45628, 271726, 82654, 627172}
> omariQ /@ {45628, 271726, 82654, 627172}
>
> {{17,34,187,374},{91},{17,34,187,374},{91}}
>
> {True,True,True,True}
>
> And this counts the 5-digit numbers that can be factored that way:
>
> Timing@Length@Select[Range[10000, 99999], omariQ]
>
> {4.5 Second, 5887}
>
> That's more than 6.5% of numbers in the range.
>
> If we want only those for which 17 can be the common factor, here they are up to 5 digits:
>
> Timing@Select[Range@99999, omariQ[#, 17] &]
>
> {1.296 Second, {41140, 45628, 82280, 82654}}
>
> 91 yields quite a few more choices:
>
> Timing@Select[Range@99999, omariQ[#, 91] &]
>
> {1.25 Second, {10010, 11102,
>      12012, 12103, 13013, 13104, 14014, 14105, 15015, 15106, 16016, 16107,
>       17017, 17108, 18018, 18109, 20020, 20111, 21021, 21203, 22113, 22204,
>       23023, 23114, 23205, 24024, 24115, 24206, 25025, 25116, 25207, 26026,
>       26117, 26208, 27027, 27118, 27209, 30030, 30121, 30212, 31031, 31122,
>       31304, 32032, 32214, 32305, 33124, 33215, 33306, 34034, 34125, 34216,
>       34307, 35035, 35126, 35217, 35308, 36036, 36127, 36218, 36309, 40040,
>       40131, 40222, 40313, 41041, 41132, 41223, 41405, 42042, 42133, 42315,
>       42406, 43043, 43225, 43316, 43407, 44135, 44226, 44317, 44408, 45045,
>       45136, 45227, 45318, 45409, 50050, 50141, 50232, 50323, 50414, 51051,
>       51142, 51233, 51324, 51506, 52052, 52143, 52234, 52416, 52507, 53053,
>       53144, 53326, 53417, 53508, 54054, 54236, 54327, 54418, 54509, 60060,
>       60151, 60242, 60333, 60424, 60515, 61061, 61152, 61243, 61334, 61425,
>       61607, 62062, 62153, 62244, 62335, 62517, 62608, 63063, 63154, 63245,
>       63427, 63518, 63609, 70070, 70161, 70252, 70343, 70434, 70525, 70616,
>       71071, 71162, 71253, 71344, 71435, 71526, 71708, 72072, 72163, 72254,
>        72345, 72436, 72618, 72709, 80080, 80171, 80262, 80353, 80444, 80535,
>        80626, 80717, 81081, 81172, 81263, 81354, 81445, 81536, 81627, 81809,
>        90090, 90181, 90272, 90363, 90454, 90545, 90636, 90727, 90818}}
>
> Bobby
>
> On Sun, 23 Jan 2005 02:02:17 -0500 (EST), F. omari <towtoo2002 at yahoo.com> wrote:
>
>>
>> i want to investigate the following two equations:
>> a * const = z
>> a_Reversed * const = z_Reversed
>> where a, z, and their reversed form and const are all positive integers
>> ie such that:
>> 2684 * 17 = 45628
>> 4862 * 17 = 82654
>> 2986 * 91 = 271726
>> 6892 * 91 = 627172
>> it happened that many multipliers of 91 have such a property.
>> while the multipliers of 17 have only 5 cases in the interval of 1 to 3000
>> the following code will investigate the multipliers of 17, to investigate another number just replace 17. and you may increase the interval of investigation. i am sure that my code is an old fashion one, please any other ideas about a more functional code.
>> a = Table[i, {i, 1, 3000}]; zR = ""; aR = 0; z = "";
>> Do[aR = ToExpression[StringReverse[ToString[a[[i]]]]];
>>         z = ToString[a[[i]]*17];
>>        zR = StringReverse[ToString[aR*17]];
>>        If[zR == z, Print[a[[i]]]], {i, 1, 3000}]
>>
>> 242
>> 484
>> 2442
>> 2662
>> 2684
>> regards
>>
>>
>>
>>
>>
>
>
>

--
DrBob at bigfoot.com
www.eclecticdreams.net

```

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