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 Greatest Root Difference (Posted on 2011-05-25)
(A) Determine the probability that for the value of a base ten positive integer M drawn at random between 11(base ten) and 10,000(base ten) inclusively, the base ten number [√M] - [3√M] is equal to the product of the digits of M.

(B) Determine the probability that for the value of a duodecimal (base 12) positive integer N drawn at random between 11(duodecimal) and 10,000(duodecimal) inclusively, the duodecimal number [√N] - [3√N] is equal to the product of the digits of N.

(C) Determine the probability that for the value of a hexadecimal (base 16) positive integer P drawn at random between 11(hexadecimal) and 10,000(hexadecimal) inclusively, the hexadecimal number [√P] - [3√P] is equal to the product of the digits of P.

Notes:

(i) None of M, N and P can contain any leading zero.

(ii) [x] denotes the greatest integer ≤ x.

 No Solution Yet Submitted by K Sengupta No Rating

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 computer solutions Comment 1 of 1

DEFDBL A-Z
FOR bse = 10 TO 16 STEP 2
PRINT bse
ct = 0: cta = 0
FOR m = bse + 1 TO bse * bse * bse * bse
cta = cta + 1
mw = m
pr = 1
WHILE mw > 0
pr = pr * (mw MOD bse)
mw = mw \ bse
WEND
diff = INT(SQR(m) + .000001) - INT(m ^ (1 / 3) + .000001)
IF pr = diff THEN
PRINT m,
mw = m
ms\$ = ""
WHILE mw > 0
ms\$ = MID\$("0123456789ABCDEF", (mw MOD bse) + 1, 1) + ms\$
mw = mw \ bse
WEND
PRINT ms\$,
PRINT pr: ct = ct + 1
END IF
NEXT
PRINT ct; cta, ct / cta, cta / ct: PRINT
DO: a\$ = INKEY\$: LOOP UNTIL a\$ > ""
NEXT

produces the following tables. Base 14 has been thrown in for evenness of presentation. In each case, the first line identifies the base (the base of course being shown in base-10); the list of base-10 values that when converted to the given base will satisfy the results, that value shown both in decimal, and in the given base and the given product (= difference), shown in decimal; and the summary line shows the count of satisfactory numbers, the total number in the range, the probability and the reciprocal of the probability. The D is part of the floating point representation--it stands for "* 10^":

`1011           11             121           21             2116          116            6132          132            6181          181            8222          222            8341          341            12531          531            151164         1164           241183         1183           243172         3172           423217         3217           424126         4126           484134         4134           484143         4143           484162         4162           484216         4216           484223         4223           486125         6125           606152         6152           606215         6215           6021  9990      2.102102102102102D-03       475.7142857142857`
`1213           11             137           31             349           41             4163          117            7172          124            8194          142            8253          191            9265          1A1            10326          232            12361          261            12493          351            15592          414            16602          422            161023         713            211167         813            241189         831            241972         1184           322413         1491           362941         1851           403985         2381           484050         2416           484059         2423           484070         2432           484105         2461           489089         5315           759133         5351           759375         5513           759397         5531           7511390        6712           8411401        6721           8414415        8413           9614437        8431           9632  20724     1.54410345493148D-03        647.625`
`1415           11             116           12             229           21             243           31             357           41             4218          118            8241          133            9268          152            10351          1B1            11365          1C1            12855          451            201194         614            241206         622            242199         B31            333006         114A           403018         1158           403185         1237           423237         1273           423553         141B           443978         1642           484033         1681           486100         231A           606109         2325           606148         2352           606217         23A1           608468         312C           728478         3138           728490         3146           728516         3164           728543         3183           728598         31C2           729283         3551           7511273        4173           8415696        5A12           10015709        5A21           10022195        8135           12030407        B11D           14330563        B1D1           14335491        CD11           15635954        D162           15636037        D1C1           15636084        D216           15636095        D223           15643  38402     1.119733347221499D-03       893.0697674418604`
`1618           12             233           21             265           41             481           51             5282          11A            10481          1E1            14497          1F1            15578          242            161361         551            254442         115A           504653         122D           524905         1329           545345         14E1           568508         213C           728521         2149           728550         2166           728596         2194           728643         21C3           7210261        2815           8010321        2851           8012786        31F2           9012831        321F           9012853        3235           9012883        3253           9014401        3841           9615153        3B31           9916685        412D           10416850        41D2           10422036        5614           12022050        5622           12022081        5641           12028970        712A           14028997        7145           14029012        7154           14029090        71A2           14029210        721A           14029221        7225           14054037        D315           19554097        D351           19562065        F271           21040  65520     6.105006105006105D-04       1638`

 Posted by Charlie on 2011-05-25 19:16:27

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