(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.
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
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Posted by Charlie
on 2011-05-25 19:16:27 |
computer solutions
