Erich Friedman sent the easiest proof.  Divide n into 1, 11, 111, 1111, 11111 ... and take the remainder each time.  By the Pigeonhole Principle, two remainders will be the same, eventually.  Subtract.

Juha Saukkola's proof :  Divide n into 1, 10, 100, 1000 ..., and take the remainder each time.  By the Pigeonhole Principle, eventually there must be a sum of remainders which add up to a multiple of n.

Willi Moehring, Scott Purdy, BDS McConnell, John Robertson, Bob Harris, Jim Cook, Joseph W. DeVincentis, and Dave Tuller all sent proofs.

Does anyone see any revalations coming out of this?   Data and program by Rick Heylen.
2 divides 10
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296 divides 111000
297 divides 1111011111111111111
298 divides 1101110
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302 divides 11100010
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310 divides 1110110
311 divides 101010001
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318 divides 1000110
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320 divides 1000000
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384 divides 1110000000
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388 divides 1110000100
389 divides 1011011
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395 divides 100100110
396 divides 11111111111111111100
397 divides 11010001
398 divides 1110000110
399 divides 1111011111
400 divides 10000
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402 divides 100100010
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418 divides 1001110
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420 divides 1010100
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422 divides 1101000110
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426 divides 100110
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429 divides 111111
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452 divides 101101100
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454 divides 110011010
455 divides 10010
456 divides 11001000
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459 divides 111101101011
460 divides 11010100
461 divides 111101
462 divides 1111110
463 divides 110100011
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467 divides 100000111
468 divides 1010111111100
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472 divides 11011111000
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478 divides 11111110
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487 divides 1100110111
488 divides 100101000
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495 divides 1111111111111111110
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499 divides 11100000011
500 divides 1000
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576 divides 111111111000000
577 divides 110001011
578 divides 11001100010
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634 divides 10000001111110
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800 divides 100000
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978 divides 1000000110
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989 divides 11111011111101001
990 divides 1111111111111111110
991 divides 110001
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1094 divides 1001010
1095 divides 11101110
1096 divides 10001000
1097 divides 1111001011
1098 divides 1110110111010
1099 divides 10010111111111111
1100 divides 1100
1101 divides 1101
1102 divides 10100111010
1103 divides 1110100011
1104 divides 100001010000
1105 divides 1001110110
1106 divides 1100011010
1107 divides 1101011100111
1108 divides 11011110100
1109 divides 1100010101101
1110 divides 1110
1111 divides 1111
1112 divides 110110101000
1113 divides 100111011
1114 divides 11101010
1115 divides 10011110010
1116 divides 1111101011100
1117 divides 11011110101
1118 divides 1100111100010
1119 divides 101011011
1120 divides 100100000
1121 divides 100100010001
1122 divides 1001110110
1123 divides 100011011
1124 divides 10100011100
1125 divides 111111111000
1126 divides 11010011110
1127 divides 110100011
1128 divides 10011000
1129 divides 100000011001
1130 divides 10110110
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1150 divides 11010100
1151 divides 110100011111
1152 divides 1111111110000000
1153 divides 100100001
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1155 divides 1111110
1156 divides 110011000100
1157 divides 11111010001
1158 divides 110010
1159 divides 10001011
1160 divides 1101101000
1161 divides 101011111101
1162 divides 101000000010
1163 divides 1001101011101
1164 divides 10011011100
1165 divides 1111100110
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1167 divides 11010100011
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1169 divides 100100101101
1170 divides 101011111110
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1172 divides 11000011100
1173 divides 1011111111111
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1178 divides 110110010110
1179 divides 1001100111111
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1184 divides 11100000
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1188 divides 111101111111111111100
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1190 divides 1001110110
1191 divides 11000101011
1192 divides 110111000
1193 divides 11011100101
1194 divides 11000101110
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1196 divides 1001010110100
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1199 divides 1111100111
1200 divides 1110000
1201 divides 10111100101
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1205 divides 1111010
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1211 divides 1011100110111
1212 divides 101111100
1213 divides 111000110111
1214 divides 10010110101010
1215 divides 1001111101110
1216 divides 11001000000
1217 divides 101011
1218 divides 110101110
1219 divides 1100000001
1220 divides 10010100
1221 divides 111111
1222 divides 11001000010
1223 divides 100111111
1224 divides 10111110111000
1225 divides 110000100
1226 divides 1100010110
1227 divides 11100010101
1228 divides 1000110001100
1229 divides 100001111001
1230 divides 100011001110
1231 divides 100001111001
1232 divides 10010000
1233 divides 10110100101111
1234 divides 11001110
1235 divides 10000000010
1236 divides 111011000100
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1238 divides 110100100110
1239 divides 111110000001
1240 divides 111011000
1241 divides 10011111011
1242 divides 1010011111110
1243 divides 100110110001
1244 divides 10101000100
1245 divides 11111000010
1246 divides 11010011110
1247 divides 1101101
1248 divides 1010100000
1249 divides 111100110001
1250 divides 10000
1251 divides 100111101111
1252 divides 1100101100
1253 divides 11010111
1254 divides 1111000110
1255 divides 1100010010
1256 divides 10101101011000
1257 divides 111110001
1258 divides 1000110
1259 divides 100001111
1260 divides 111101111100
1261 divides 1110110001
1262 divides 1010111110
1263 divides 101010101001
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1265 divides 111110010
1266 divides 1100010100110
1267 divides 101011110011
1268 divides 100000011111100
1269 divides 111101101011
1270 divides 110100110
1271 divides 1001001001001
1272 divides 100011000
1273 divides 101111000001
1274 divides 101111010
1275 divides 10001100
1276 divides 11110100100
1277 divides 1011010111001
1278 divides 1001110111110
1279 divides 1001010001011
1280 divides 100000000
1281 divides 110010001101
1282 divides 11100011110
1283 divides 1111010001
1284 divides 11010011100
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1287 divides 111111111111111111
1288 divides 11101111000
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1291 divides 10111010011
1292 divides 1101010100
1293 divides 101010100011
1294 divides 101100110010
1295 divides 101010
1296 divides 11111111010000
1297 divides 11000000011111
1298 divides 1001011110
1299 divides 10001001
1300 divides 100100
1301 divides 11001010111
1302 divides 100000110
1303 divides 1101110101011
1304 divides 1010111000
1305 divides 101111011110
1306 divides 111010
1307 divides 110110101001
1308 divides 100101011100
1309 divides 100111011
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1312 divides 1111100000
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1335 divides 10111111110
1336 divides 11111011000
1337 divides 11101111
1338 divides 10011110010
1339 divides 1110110001
1340 divides 110101100
1341 divides 10111101111
1342 divides 11011110
1343 divides 110001101
1344 divides 10101000000
1345 divides 1001000110
1346 divides 1001000010
1347 divides 1101110111001
1348 divides 101100
1349 divides 101111110011
1350 divides 110111111100
1351 divides 101011110011
1352 divides 10111101000
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1354 divides 10010101101010
1355 divides 111110
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1365 divides 101010
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1368 divides 1111011111000
1369 divides 10100111001
1370 divides 100010
1371 divides 101101010001
1372 divides 110000100
1373 divides 100111110101
1374 divides 111101111010
1375 divides 11000
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1378 divides 1001110110
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1425 divides 1100100
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1427 divides 11011111101101
1428 divides 10011101100
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1430 divides 10010
1431 divides 11111101101
1432 divides 11010111000
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1434 divides 1010000111110110
1435 divides 10111010
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1438 divides 1010000000010
1439 divides 110010111
1440 divides 11111111100000
1441 divides 1111011
1442 divides 11101100010
1443 divides 10101
1444 divides 1110111100
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1450 divides 110110100
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1452 divides 100011110100
1453 divides 1000011001101
1454 divides 1011000100010
1455 divides 1001101110
1456 divides 10010000
1457 divides 1101110110101
1458 divides 1011111110010
1459 divides 10010110001
1460 divides 1000100
1461 divides 11111111001
1462 divides 11010110010
1463 divides 1000000001
1464 divides 100101000
1465 divides 1100001110
1466 divides 100011011110
1467 divides 1010100111111
1468 divides 110100
1469 divides 1101100010101
1470 divides 11000010
1471 divides 100111010001
1472 divides 110101000000
1473 divides 100101010101
1474 divides 1101100110
1475 divides 1101111100
1476 divides 11100111101100
1477 divides 110100011
1478 divides 1101110
1479 divides 111101001
1480 divides 111000
1481 divides 10010110101
1482 divides 101101010010
1483 divides 11010010001
1484 divides 10011101100
1485 divides 11110111111111111110
1486 divides 101100010
1487 divides 101010011101
1488 divides 100000110000
1489 divides 111010011111
1490 divides 1101110
1491 divides 100101000111
1492 divides 10101101100
1493 divides 10100100101011
1494 divides 1001101111110
1495 divides 100101011010
1496 divides 1001011000
1497 divides 1110100100001
1498 divides 110111001110110
1499 divides 110010111
1500 divides 111000
1501 divides 1101000011
1502 divides 11110001110
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1505 divides 11010011110
1506 divides 11100101010
1507 divides 110011
1508 divides 1110001100
1509 divides 11010100101
1510 divides 11100010
1511 divides 10110101
1512 divides 1010111010111000
1513 divides 11010101
1514 divides 1011010010110
1515 divides 10111110
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1517 divides 10001100111
1518 divides 111110010
1519 divides 10010011011
1520 divides 110010000
1521 divides 111110101011
1522 divides 110111000110
1523 divides 10110111101
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1527 divides 100101100011
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1529 divides 1010010001
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1538 divides 1000110100010
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1544 divides 11001000
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1547 divides 100111011
1548 divides 1111101110100
1549 divides 10000110101
1550 divides 11101100
1551 divides 1001110011
1552 divides 111000010000
1553 divides 1000011001111
1554 divides 101010
1555 divides 1010100010
1556 divides 101101100
1557 divides 1111111011
1558 divides 10110010010
1559 divides 1101011011
1560 divides 10101000
1561 divides 1010000001101
1562 divides 10011100110
1563 divides 1011100011
1564 divides 11011100010100
1565 divides 110010110
1566 divides 10110111010110
1567 divides 1000111111
1568 divides 110000100000
1569 divides 1000100101101
1570 divides 101011010110
1571 divides 100001111101
1572 divides 10100100
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1575 divides 111101111100
1576 divides 1101000101000
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1580 divides 1001001100
1581 divides 10010011100001
1582 divides 1100111101110
1583 divides 1100101101
1584 divides 1111111111111111110000
1585 divides 10000001111110
1586 divides 10100100010
1587 divides 1011111111111
1588 divides 1101000100
1589 divides 1011100100101
1590 divides 1000110
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1592 divides 111000011000
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1594 divides 100011011010
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1600 divides 1000000
1601 divides 111111001
1602 divides 10111111110
1603 divides 1110111110101
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1606 divides 1100110
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1608 divides 10010001000
1609 divides 10001010100011
1610 divides 111011110
1611 divides 111110011101
1612 divides 11100111100
1613 divides 1001001100011
1614 divides 10101001110
1615 divides 110101010
1616 divides 1010000
1617 divides 10111101
1618 divides 111010001110
1619 divides 11111001101
1620 divides 111111110100
1621 divides 100001111
1622 divides 10010100010
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1624 divides 11010111000
1625 divides 1001000
1626 divides 1111111111111110
1627 divides 1010000100111
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1640 divides 11111000
1641 divides 100101
1642 divides 100001010110
1643 divides 1100000001
1644 divides 111011100
1645 divides 111100010
1646 divides 11000111010
1647 divides 111110100111
1648 divides 111000010000
1649 divides 101001001001
1650 divides 11111100
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1652 divides 1110110010100
1653 divides 1010011101
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1655 divides 1110111110
1656 divides 11011111011000
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1660 divides 10101100
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1668 divides 11011010100
1669 divides 110001001101
1670 divides 111110110
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1684 divides 10011001100
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1688 divides 110100011000
1689 divides 100011110001
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1696 divides 10001100000
1697 divides 11000010001
1698 divides 100111100010
1699 divides 11010011011
1700 divides 1110100
1701 divides 11010111010101
1702 divides 100001010
1703 divides 1000000011101
1704 divides 10011000
1705 divides 110010010
1706 divides 10010100010
1707 divides 1011010011
1708 divides 101001000100
1709 divides 1011000101011
1710 divides 11110111110
1711 divides 101011111011
1712 divides 1000100110000
1713 divides 1000010001
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1716 divides 11111100
1717 divides 1001011
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1719 divides 101111011011
1720 divides 1101101000
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1722 divides 1100101000110
1723 divides 111111101
1724 divides 10110010100
1725 divides 1000010100
1726 divides 1100011101010
1727 divides 1001110111111
1728 divides 1101111111000000
1729 divides 1000000001
1730 divides 10110011010
1731 divides 110111110101
1732 divides 1000100100
1733 divides 10111010001
1734 divides 1010100110010
1735 divides 10101111010
1736 divides 10000011000
1737 divides 110011111101
1738 divides 100100110
1739 divides 100111011111
1740 divides 1101011100
1741 divides 1110011111
1742 divides 1110011110
1743 divides 10100000001
1744 divides 10010101110000
1745 divides 11001000010
1746 divides 1100111011110
1747 divides 110000000001
1748 divides 1110111100
1749 divides 100111011
1750 divides 1001000
1751 divides 100001100101
1752 divides 1110111000
1753 divides 100100011101
1754 divides 10101110100110
1755 divides 10110010111110
1756 divides 1000110001100
1757 divides 100001101011
1758 divides 10011010110
1759 divides 11101100011
1760 divides 1100000
1761 divides 1101011101011
1762 divides 1001001010
1763 divides 1010010001111
1764 divides 1111111010100
1765 divides 101011010010
1766 divides 11011010
1767 divides 1001101100001
1768 divides 100111011000
1769 divides 10100101111111
1770 divides 1001011110
1771 divides 10010101101
1772 divides 11000000100
1773 divides 101111011011
1774 divides 110000001110
1775 divides 1001100
1776 divides 1110000
1777 divides 101010011
1778 divides 10011100111110
1779 divides 10010111001
1780 divides 1101010100
1781 divides 10011001
1782 divides 11111111111111110110
1783 divides 1100111
1784 divides 1001111001000
1785 divides 1001110110
1786 divides 1000011110110
1787 divides 1001001011111
1788 divides 1000011101100
1789 divides 11101101011
1790 divides 110101110
1791 divides 10101110011011
1792 divides 100100000000
1793 divides 100100011011
1794 divides 100101011010
1795 divides 10010101110
1796 divides 11110011100
1797 divides 110000000001
1798 divides 111011110010010
1799 divides 100111110001
1800 divides 111111111000
1801 divides 11011010011011
1802 divides 1000110
1803 divides 11100110001
1804 divides 111111111100
1805 divides 111011110
1806 divides 11100010110
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1810 divides 10011110
1811 divides 100100111101
1812 divides 1000101101100
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1815 divides 10001111010
1816 divides 11001101000
1817 divides 1111010101
1818 divides 10111111111111111110
1819 divides 101001010011
1820 divides 100100
1821 divides 1110010000101
1822 divides 1010000110010
1823 divides 1110001001
1824 divides 1100100000
1825 divides 1000100
1826 divides 11111000010
1827 divides 101111100111
1828 divides 100110100100
1829 divides 1010100001
1830 divides 1001010
1831 divides 1100010000001
1832 divides 10000100011000
1833 divides 110100001011
1834 divides 101100011110
1835 divides 11010
1836 divides 11110110101100
1837 divides 1100001111
1838 divides 10101110100110
1839 divides 10110011001
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1841 divides 101100100101
1842 divides 10111110001110
1843 divides 1101101100101
1844 divides 11110100
1845 divides 1110011110110
1846 divides 101001001010
1847 divides 101101110011
1848 divides 111111000
1849 divides 10110001101111
1850 divides 11100
1851 divides 1010000001
1852 divides 11010001100
1853 divides 1101111001001
1854 divides 1111011011010
1855 divides 1001110110
1856 divides 1101101000000
1857 divides 11010010011
1858 divides 11000001010
1859 divides 10111101
1860 divides 1000001100
1861 divides 100111101111
1862 divides 1010011000110
1863 divides 1101011011011
1864 divides 111110011000
1865 divides 1010110110
1866 divides 111011111010
1867 divides 1110111111001
1868 divides 10000011100
1869 divides 101001001101
1870 divides 10010110
1871 divides 110101111101
1872 divides 101011111110000
1873 divides 11000011001
1874 divides 101101110010
1875 divides 1110000
1876 divides 110010000100
1877 divides 110110100001
1878 divides 11000000010
1879 divides 101011111011
1880 divides 10011000
1881 divides 111111111111111111
1882 divides 1000010110
1883 divides 10110011001111
1884 divides 11011110110100
1885 divides 111000110
1886 divides 10010011010
1887 divides 100011
1888 divides 1101111100000
1889 divides 110000110101111
1890 divides 10101110101110
1891 divides 1001011101111
1892 divides 11111101100
1893 divides 1111101001101
1894 divides 10110011010
1895 divides 1001011010
1896 divides 101010111000
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1900 divides 1100100
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1907 divides 11011100001
1908 divides 1111110110100
1909 divides 1001001100011
1910 divides 111011110
1911 divides 10111101
1912 divides 1111111000
1913 divides 10010111101
1914 divides 1111010010
1915 divides 110101010
1916 divides 10011100
1917 divides 111001111101
1918 divides 100110010
1919 divides 1111101
1920 divides 1110000000
1921 divides 1111011101111
1922 divides 10100110
1923 divides 1011101010111
1924 divides 1010100
1925 divides 100100
1926 divides 1110101110110
1927 divides 10100011101011
1928 divides 111101000
1929 divides 1110110100111
1930 divides 110010
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1932 divides 10110011100
1933 divides 101010011011
1934 divides 100100001010
1935 divides 111110111010
1936 divides 110110000
1937 divides 101011011101
1938 divides 1111011010110
1939 divides 101101011101111
1940 divides 1110000100
1941 divides 10110011001
1942 divides 10011010
1943 divides 10111101101111
1944 divides 100111110111000
1945 divides 10110110
1946 divides 10100100010
1947 divides 100101111
1948 divides 110011011100
1949 divides 1100111101
1950 divides 1010100
1951 divides 1101001101001
1952 divides 10010100000
1953 divides 101010111111
1954 divides 10111100010
1955 divides 1101110001010
1956 divides 10000001100
1957 divides 10010100011
1958 divides 11001111110
1959 divides 10010001110001
1960 divides 1100001000
1961 divides 100011
1962 divides 11111110110
1963 divides 1111111001
1964 divides 10010101010100
1965 divides 1010010
1966 divides 110001111010
1967 divides 101101111111
1968 divides 100011001110000
1969 divides 10000110001101
1970 divides 11010001010
1971 divides 101111001111
1972 divides 11110100100
1973 divides 1111010111
1974 divides 1001010001110
1975 divides 1001001100
1976 divides 1000000001000
1977 divides 1001111111001
1978 divides 111110111111010010
1979 divides 10000011101111
1980 divides 11111111111111111100
1981 divides 10000010010011
1982 divides 1100010
1983 divides 111011011101
1984 divides 111011000000
1985 divides 110100010
1986 divides 10110001110
1987 divides 101000110111
1988 divides 1100010100
1989 divides 1001101111101
1990 divides 1110000110
1991 divides 11110001001
1992 divides 1111100001000
1993 divides 1110101
1994 divides 1111100000010
1995 divides 11110111110
1996 divides 1110000001100
1997 divides 1000001011101
1998 divides 1111111111111111111111111110
1999 divides 1111000000111

#include <stdio.h>
#define NUM 2000

int main(int argc, char* argv[]){
 signed long pow[NUM],val[NUM],x,num,ten;
 int i,j,count;
 for(num=2;num<NUM;num++){
 for(i=0;i<NUM;pow[i++]=0);
 count=0;
 for(ten=1,x=1;x<NUM;x++){
 val[x]=ten;

for(j=0;j<NUM;j++)if(pow[j]&&!pow[(j+ten)%num]&&pow[j]!=x)pow[(j+ten)%num]=x;
 if(!pow[ten])pow[ten]=x;
 ten=(10*ten)%num;
 if(pow[0])break;
 }

//for(i=0;i<=num;i++)printf("pow[%d]=%ld\tval[%d]=%ld\n",i,pow[i],i,val[i]);
 x=num;
 printf("%ld\tdivides\t",x=num);
 if(pow[0]){
 while(x){
 while(--count>pow[x%num]-1)printf("0");
 count=pow[x%num]-1;
 printf("1");
 x=(num+x-val[pow[x%num]])%num; }
 while(count-->0)printf("0");
 }
    else {
    printf("Can't do it!");
    getchar();
    }
 printf("\n");
   // getchar();
 }
}