All of the digits of a three-digit integer are distinct and non-zero. Furthermore, the three-digit integer is divisible by 19. Find the largest three-digit integer that has these properties.
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To find the largest three-digit integer with distinct and non-zero digits that is divisible by 19, we can start with the largest possible three-digit number and work backwards.
The largest three-digit number is 999, but this number is not divisible by 19. So, we need to find the next largest three-digit number.
To find this number, we can begin by listing the three-digit numbers in descending order:
998, 997, 996, 995, ..., 100
We then check each of these numbers to see if they are divisible by 19.
Starting with 998, we can divide it by 19 to see if it is divisible evenly:
998 รท 19 = 52 remainder 10
Since there is a remainder of 10, 998 is not divisible by 19. We continue to check the remaining numbers.
997 รท 19 = 52 remainder 9
996 รท 19 = 52 remainder 8
995 รท 19 = 52 remainder 7
994 รท 19 = 52 remainder 6
993 รท 19 = 52 remainder 5
992 รท 19 = 52 remainder 4
991 รท 19 = 52 remainder 3
990 รท 19 = 52 remainder 2
989 รท 19 = 52 remainder 1
988 รท 19 = 52 remainder 0
We have found a number, 988, that is divisible by 19. However, this number does not have distinct digits.
Continuing our search, we can try the next largest number, 987, and check if it is divisible by 19.
987 รท 19 = 51 remainder 18
Since there is a remainder of 18, 987 is not divisible by 19.
We continue this process, checking each number until we find a three-digit number that is divisible by 19.
986 รท 19 = 51 remainder 17
985 รท 19 = 51 remainder 16
984 รท 19 = 51 remainder 15
983 รท 19 = 51 remainder 14
982 รท 19 = 51 remainder 13
981 รท 19 = 51 remainder 12
980 รท 19 = 51 remainder 11
979 รท 19 = 51 remainder 10
978 รท 19 = 51 remainder 9
977 รท 19 = 51 remainder 8
976 รท 19 = 51 remainder 7
975 รท 19 = 51 remainder 6
974 รท 19 = 51 remainder 5
973 รท 19 = 51 remainder 4
972 รท 19 = 51 remainder 3
971 รท 19 = 51 remainder 2
970 รท 19 = 51 remainder 1
969 รท 19 = 51 remainder 0
We have found a number, 969, that is divisible by 19 and has distinct digits.
Therefore, the largest three-digit integer with distinct and non-zero digits that is divisible by 19 is 969.