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

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MC II-B Number Theory Question 1.6(b) and 1.8
by Jeff Ho - Friday, August 26, 2022, 11:54 AM

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.

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Re: MC II-B Number Theory Question 1.6(b) and 1.8
by John Lensmire - Monday, August 29, 2022, 11:10 AM

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 $z$?