please help explain why 2000 is fortunate We will call a num

please help explain why 2000 is fortunate

We will call a number N fortunate if it can be written with four equal digits in some base b epsilon Z+. Clearly 2222 is fortunate, as is 3333_6. Why is 2000 fortunate?

Solution

2000 is fortunate because in base 7 it can be written as 5555

Here is an example to make things clear.

Let x=45x=45 and b=3b=3.

We see that 45=1200345=12003. Read up the last column to get the base-3 expansion you seek. Let us check.

133+232+0+0=27+18=45.133+232+0+0=27+18=45.

I hope this helps you.