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 Divisible by 11,111 (Posted on 2008-10-02)
How many positive 10-digit integers of the form ABCDEFGHIJ, with non leading zeroes and each letter representing a different base 10 digit from 0 to 9, are divisible by 11,111 ?

Note : Try to solve this problem analytically, although computer program/ spreadsheet solutions are welcome.

 See The Solution Submitted by K Sengupta Rating: 2.6667 (3 votes)

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 computer solution (spoiler) | Comment 2 of 6 |

The program listed below checks all multiples of 11,111 over 1,000,000,000 and under 10,000,000,000 to see if no digit is repeated in it. In practice it starts with 1,023,467,543, the first multiple of 11,111 within the range, and ends with the highest all-digit number, 9876543210.  It finds 3456 such multiples of 11,111.

As there are about 900,000,000/11,111 multiples of 11,111, and 9!/10^9 of these would be expected, at random, to have all the digits without repetition, one would expect only about 29 integers filling the description.  The excess must have to do with digital sums and other divisibility rules. 11,111 is factored as 41 * 271.

The list begins:

1023489765
1023589764
1024389756
1024689753
1025389746
1025689743
1026489735
1026589734
1032489675
1032589674
1034289657
1034789652
1035289647
1035789642
1037489625
1037589624
1042389576
1042689573
1043289567
1043789562
1046289537
1046789532
1047389526
1047689523
1052389476
1052689473
1053289467
1053789462
1056289437
1056789432
1057389426
1057689423
1062489375
1062589374
1064289357
1064789352
1065289347
1065789342
1067489325
1067589324

...

and ends with

...

9823501764
9824301756
9824601753
9825301746
9825601743
9826401735
9826501734
9832401675
9832501674
9834201657
9834701652
9835201647
9835701642
9837401625
9837501624
9842301576
9842601573
9843201567
9843701562
9846201537
9846701532
9847301526
9847601523
9852301476
9852601473
9853201467
9853701462
9856201437
9856701432
9857301426
9857601423
9862401375
9862501374
9864201357
9864701352
9865201347
9865701342
9867401325
9867501324
9873401265
9873501264
9874301256
9874601253
9875301246
9875601243
9876401235
9876501234

DEFDBL A-Z
CLS
n = 1023467543
DO
REDIM used(9)
ns\$ = LTRIM\$(STR\$(n))
good = 1
FOR i = 1 TO 10
v = VAL(MID\$(ns\$, i, 1))
IF used(v) THEN good = 0: EXIT FOR
used(v) = 1
NEXT
IF good THEN PRINT n: ct = ct + 1: IF ct = 40 THEN DO: LOOP UNTIL INKEY\$ > ""
n = n + 11111
LOOP UNTIL n > 9876543210#
PRINT ct

 Posted by Charlie on 2008-10-02 12:40:46

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