For 8.19, this seems to be a typo. The remainders of $2, 4, 8, 16\dots$, are $2, 4, 1, 2, \dots$. (It should be fixed now. Thanks for pointing it out!)

The pattern of the powers of $3$ repeats every $4$ exponents. When we divide (the exponent) $2015$ by $4$, we look at the remainder to see which of the four possible remainders, $3,9,7,1$ corresponds to $3^{2015}$. Since we get remainder $3$ when dividing by $4$, the third remainder is the one we are looking for, that is, $7$.

For 8.30: Two numbers have the same remainder $R$ when dividing by $D$ if their difference is divisible by the number $D$. So, for example, the number $D$ must divide $571 - 513 = 58$. Look at the other two possible differences of two of the numbers to find clues as to what $D$ can be. Make sure to look at the prime factors that divide each of the differences.