Question

Show that for any natural number n there is a number composed of digits 5 and 0 only and divisible by n

Answer 1

Answer:

Consider the numbers 1,11,111,...,111...11 (1 is repeated n+1 times). Since these are n+1 numbers we can use the pegionhole principle to deduce that 2 of them are congruent modulo n. Find the absolute value of the difference between the 2 numbers that are congruent modulo n. The difference is made up of 1(s) and zeros. This number is also divisible by n. (done)

In general if n is indivisible by 2 and 5, there exists a multiple of n that is written as repeated sequences of digits. For example n has a multiple that is in the form 249249249...249 The proof is very similar to the previous argument

I hope this will help uhh too much