## 1998 multiple

1998 = 2 . 999 = 2 . 3^{3} . 37. A number formed only by the digits 0 and 3 is a multiple of 3^{3} if and only if the number of digits 3 is a multiple of 9 (because by dividing it by 3 we get a number that has only the digits 0 and 1 which must be a multiple of 9, which happens if and only if the number of digits 1 is a multiple of 9).

Thus, the desired number must be at least 9 digits 3, and must end with 0 because it is even. The smallest number with these properties is 3333333330, which is a multiple of 1998 because it is even, is a multiple of 3^{3} and is a multiple of 37 because it is a multiple of 111 (equals 111,300,300,330).

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