## Online Course Discussion Forum

### MC II-B Number Theory Question 1.6(b) and 1.8

Due to the self-paced course ending at question 1.6(a), I still wasn't able to figure out questions 1.6b and 1.8 from the example questions.

Hi Jeff.

Do you have the textbook to go along with the course? The textbook includes answers and solution ideas to the example problems as well, so that can be a good reference.

1.6(b) should be a good one to practice what was done in 1.6(a). Did the procedure in that question make sense? 1.6(b) will follow the same pattern of

- Multiplying by 2 and recording the ones digit as the first decimal.
- Removing the ones digit and repeating the procedure.

For example, to start, $$0.\overline{3} \times 2= 0.3333\cdots \times 2 = 0.6666\cdots = 0.\overline{6}$$ so the first decimal place of $\dfrac{1}{3}$ written in binary is $0$. Then you multiply $0.\overline{6}$ by two and the new ones place is the next digit in the decimal, etc.

One hint: Here the procedure will never stop, but a pattern should show up (pretty quickly) in the decimals!

Problem 1.8 is a little more of a puzzle, so let me give a bigger hint. Note converting from increasing bases to decimal gives:

- $111_2 = 7_{10}$
- $21_3 = 7_{10}$
- $12_5 = 7_{10}$
- $10_7 = 7_{10}$

Following this type of pattern, what could we get for $x$, $y$, and $z$?

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