Question

Let there be a sequence of integers where a_1 is less than 2^101. For each a_i, multiply a_i by 5^100 in base 10, replace each digit with its remainder mod 2, and read that number in binary. Switch it to base 10 and call this number a_(i+1). Show that for any a_1 less than 2^101, the sequence is periodic and has period 2^n for some n.

Answer 1

what do you mean by "switch it to base 10"?
Do you mean read that string of 1's and 0's as a base-10 number,
or convert it to base 10?

I suggest you start with a smaller exponent, and see what happens if it is 2,3,4 etc.
5^100 is very large to work with.

Answer 2

I mean that you read it in base 2 and convert it to base 10. I looked at small exponents and I’m stuck. Any help please and thank you.