I can't understand this solution with $\equiv$

Question

I found a solution for finding the units digit of a sum. It said that the $\equiv$ means "has the same units digit as". Using this, they said that $3^1+3^3+3^5+...+3^{2009} \equiv 3+7+3+...+3+7+3$ But they said that this is $\equiv$ to $0+0+0+...+3$. How did they come up with this?

Answer 1

You correctly identified that $\pmod{10}$, the sequence is equivalent to

$$3 + 7 + 3 + 7 + 3 + 7 + 3 + 7 + \cdots.$$

You then asked how it could be concluded that the final sum is equivalent to $3 \pmod{10}.$

$(2009 - 1)$ is equivalent to $(0) \pmod{4}$, rather than $(2) \pmod{4}$. Therefore, there is an odd number of terms in the sequence, rather than an even number of terms.

If there were an even number of terms, the sum of the sequence could be viewed as

$$\left[ ~(3 + 7) + (3 + 7) + \cdots + (3 + 7) ~\right] \pmod{10}.$$

Since the sum $\pmod{10}$ of each pair of terms is $0$, the overall sum of the sequence would be $(0)$, if there were an even number of terms.

However, there is an odd number of terms. Therefore, the sequence should be viewed as

$$\left\{ ~\left[ ~(3 + 7) + (3 + 7) + \cdots + (3 + 7) ~\right] + 3 ~\right\} \pmod{10}.$$