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Read chapters 15 and 16 of Gallian's book three times:
- First time as if you were reading a novel - quickly and without too much attention to detail, just to learn what the main keywords and concepts and goals are.
- Second time like you were studying for an exam on the subject - slowly and not skipping anything, verifying every little detail.
- And then a third time, again at a quicker pace, to remind yourself of the bigger picture all those little details are there to paint.
Solve problems 19, 20, 30, 35, 39, S50, 53, S54, 62 in Chapter 15 of Gallian's book and problems 1, 2, 3, 9, 10, S11, 12, S13, S15 in Chapter 16 of the same book, but submit only the solutions of the problems marked with the letter "S".
This assignment is due in class on Wednesday February 6, 2008.
Just for Fun
- Take a large integer and write it in base 10. Cut away the "singles" digit, double it and subtract the result from the remaining digits. Repeat the process until the number you have left is small. Prove that the number you started from is divisible by 7 iff the resulting number is divisible by 7. Thus the example on the right shows that 86415 is divisible by 7 as 0 is divisible by 7. (I learned this trick from Bradford Hovinen).
- Find a similar criterion for divisibility by 17 and for all other divisibilities and indivisibilities.
- Note that the word "indivisibilities" has the largest number of repetitions of a single letter among all words in the English language (7 i's). I've known this fact for years but this exercise is the first time that I'm finding a semi-legitimate use for that word! (It is tied with the word honorificabilitudinitatibus for seven 'i's. You can read more about it here: http://en.wikipedia.org/wiki/Honorificabilitudinitatibus)
Here is one set of solutions by LeoDaVinci