06-240/Homework Assignment 4

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• Read sections 1.5 through 1.7 in our textbook. Remember that reading math isn't like reading a novel! If you read a novel and miss a few details most likely you'll still understand the novel. But if you miss a few details in a math text, often you'll miss everything that follows. So reading math takes reading and rereading and rerereading and a lot of thought about what you've read.

• Solve problems 1-5 on pages 32-34, but submit only the last part of each problem.

• Solve problems 6-10 on page 34, but submit only your solution of problem 10.

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 a few days ago from Bradford Hovinen). Find a similar criterion for divisibility by 17 and for all other divisibilities. Note that the word "divisibilities" has the largest number of repeatitions of a single letter among all words in the English language (6 i's). I've known this fact for years but this is the first time that I'm finding a semi-legitimate use for that word!

86415 10 ---- 8631 2 --- 861 2 -- 84 8 - 0