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# number theory

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75
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How many ways are there to fill in the tens digit and hundreds digit of 1 _ _ 4 so that it is divisible by 7?

Nov 29, 2020

#1
+2533
+1

1008 1015 1022 1029 1036 1043 1050 1057 1064 1071 1078 1085 1092 1099 1106 1113 1120 1127 1134 1141 1148 1155 1162 1169 1176 1183 1190 1197 1204 1211 1218 1225 1232 1239 1246 1253 1260 1267 1274 1281 1288 1295 1302 1309 1316 1323 1330 1337 1344 1351 1358 1365 1372 1379 1386 1393 1400 1407 1414 1421 1428 1435 1442 1449 1456 1463 1470 1477 1484 1491 1498 1505 1512 1519 1526 1533 1540 1547 1554 1561 1568 1575 1582 1589 1596 1603 1610 1617 1624 1631 1638 1645 1652 1659 1666 1673 1680 1687 1694 1701 1708 1715 1722 1729 1736 1743 1750 1757 1764 1771 1778 1785 1792 1799 1806 1813 1820 1827 1834 1841 1848 1855 1862 1869 1876 1883 1890 1897 1904 1911 1918 1925 1932 1939 1946 1953 1960 1967 1974 1981 1988

Nov 29, 2020
#2
+2533
+1

I was writing a long good answer but the page got updated. The above are all the divisors of 7 that you need. If you start counting from 1064 every 11th 7 divisible will be the one that is required. Pay attention to that pattern since it will give you a simpler way to solve the question.

1988-1008 = 980

980/7 = 140

140 - 8 = 132

so it will start from 1st number required meaning +1

132/11 = 12