# To what number b/w 0 and 4 inclusive

## Main Question or Discussion Point

Q1) To what number b/w 0 and 4 inclusive is the sum 1+2+2^2+...+2^19 congruent mod 5?

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I am reading the book What is mathematics and I am stuck on this question. If anybody could help me understand better, they would be very kind. Thankyou.

One thing that might help is this:
$$2^0 \equiv 1 \pmod 5$$
$$2^1 \equiv 2 \pmod 5$$
$$2^2 \equiv -1 \pmod 5$$
$$2^3 \equiv -2 \pmod 5$$
$$2^4 \equiv 1 \pmod 5$$
$$2^5 \equiv 2 \pmod 5$$
$$2^6 \equiv -1 \pmod 5$$
$$2^7 \equiv -2 \pmod 5$$
see a pattern?

You should also know something about a nice way to simplify $$2^0 + 2^1 + \cdots + 2^n$$

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oh ok . i see now. then the answer is 0 to that.
Thank You.