Problem - 4208
Let $\mathbb{S}$ be a set containing all the integers created by digits $1$, $2$, $\cdots$, $7$. Each digit can be used once and only once. Show that no element in $\mathbb{S}$ is a multiple of the other.
If this claim is not true, let two numbers $a, b\in\mathbb{S}$ and $a=bc$ where $c$ is a positive integer $c > 1$. Let $S(n)$ be the sum of digits of $n$, then $$S(a)\equiv S(b)\equiv 1+2+\cdots + 7\equiv 1\pmod{9}$$
Now we have $$a=bc\implies 1\equiv1\times c\pmod{9}\implies c\equiv 1\pmod{9}$$
Because $c > 1$, then $c \ge 10$. However this cannot be held since both $b$ and $c$ are seven-digit numbers.