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Greatest Root Difference (Posted on 2011-05-25) Difficulty: 3 of 5
(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    
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Solution 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^":

10
11           11             1
21           21             2
116          116            6
132          132            6
181          181            8
222          222            8
341          341            12
531          531            15
1164         1164           24
1183         1183           24
3172         3172           42
3217         3217           42
4126         4126           48
4134         4134           48
4143         4143           48
4162         4162           48
4216         4216           48
4223         4223           48
6125         6125           60
6152         6152           60
6215         6215           60
21  9990      2.102102102102102D-03       475.7142857142857
12
13           11             1
37           31             3
49           41             4
163          117            7
172          124            8
194          142            8
253          191            9
265          1A1            10
326          232            12
361          261            12
493          351            15
592          414            16
602          422            16
1023         713            21
1167         813            24
1189         831            24
1972         1184           32
2413         1491           36
2941         1851           40
3985         2381           48
4050         2416           48
4059         2423           48
4070         2432           48
4105         2461           48
9089         5315           75
9133         5351           75
9375         5513           75
9397         5531           75
11390        6712           84
11401        6721           84
14415        8413           96
14437        8431           96
32  20724     1.54410345493148D-03        647.625
14
15           11             1
16           12             2
29           21             2
43           31             3
57           41             4
218          118            8
241          133            9
268          152            10
351          1B1            11
365          1C1            12
855          451            20
1194         614            24
1206         622            24
2199         B31            33
3006         114A           40
3018         1158           40
3185         1237           42
3237         1273           42
3553         141B           44
3978         1642           48
4033         1681           48
6100         231A           60
6109         2325           60
6148         2352           60
6217         23A1           60
8468         312C           72
8478         3138           72
8490         3146           72
8516         3164           72
8543         3183           72
8598         31C2           72
9283         3551           75
11273        4173           84
15696        5A12           100
15709        5A21           100
22195        8135           120
30407        B11D           143
30563        B1D1           143
35491        CD11           156
35954        D162           156
36037        D1C1           156
36084        D216           156
36095        D223           156
43  38402     1.119733347221499D-03       893.0697674418604
16
18           12             2
33           21             2
65           41             4
81           51             5
282          11A            10
481          1E1            14
497          1F1            15
578          242            16
1361         551            25
4442         115A           50
4653         122D           52
4905         1329           54
5345         14E1           56
8508         213C           72
8521         2149           72
8550         2166           72
8596         2194           72
8643         21C3           72
10261        2815           80
10321        2851           80
12786        31F2           90
12831        321F           90
12853        3235           90
12883        3253           90
14401        3841           96
15153        3B31           99
16685        412D           104
16850        41D2           104
22036        5614           120
22050        5622           120
22081        5641           120
28970        712A           140
28997        7145           140
29012        7154           140
29090        71A2           140
29210        721A           140
29221        7225           140
54037        D315           195
54097        D351           195
62065        F271           210
40  65520     6.105006105006105D-04       1638

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