Online Course Discussion Forum
Number Theory I-B
The problems I don't know how to solve in Chapter 8 are:
8.19: I understand you divide the powers of 2 by 7 until you find a pattern, but the remainder for 2^1 is 1, the remainder for 2^2 is 3, and so on. However, in the answer key, it says it is 1, 2, 4, and so on. Is my way a calculation error ?
8.24: I understand the first part of the problem, where the pattern of the units' digits is 3, 9, 7, 1....since it is a cycle of 4, you do 2015/4= 503 remainder 3. What does that have to do with the units' digit being 7?
8.30: I just don't know how to solve the problem.
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.
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