Natural number divisible by $42$?

Question

There is a natural number divisible by $42$. The sum of digits which do not take part in the written number is $25$. Prove that there are two identical numerals in the natural number.

Answer 1

Hint: The sum of the digits of the number has to be a multiple of $3$, because it is divisible by $3$. What is the sum of all the digits there are? If every digit in the number is used only once, what is the sum of its digits?

Answer 2

Suppose that every digit is used only once then the sum of digit will be $20$(since $45-25$).
We know that $S(n)\equiv n\pmod 3$ and since the number is divisible by 3 then so $20$ should but since it's not then a non zero should be used at least twice .

Answer 3

As the sum of all digits is $45$, the sum of used digits is $20$. Thus if there were no digit with multiple occurance, the digit sum of the given number $N$ would be equal to $20$. On the other hand, we are given that $N$ is a multiple of $3$, hence the digit sum must be a multiple of $3$. As $20$ is not a multiple of $3$, this is impossible.