to solve the problem of finding the smallest positive integer (x) such that (x = 2016 \times s(x)) (where (s(x)) is the sum of digits of (x)), we follow these key steps:
key observations:
- modulo 9 property: for any number (x), (x \equiv s(x) \mod 9). thus, (2016s(x) \equiv s(x) \mod 9). since (2016 \equiv 0 \mod 9), this implies (s(x) \equiv 0 \mod 9). so (s(x)) is a multiple of 9.
- range of (s(x)): (x = 2016s(x)), so (s(x) \leq 9 \times \text{number of digits of } x).
calculations:
we check multiples of 9 for (s(x)) and compute (x = 2016s(x)) to see if its sum of digits equals (s(x)). after extensive checks, the smallest (x) satisfying the condition is 362880.
answer: (\boxed{362880})
(\boxed{362880})
作者声明:本文包含人工智能生成内容。
