SOAR Homework Seven

These homework problems are meant to expand your understanding of what goes on during class. Any you turn in will be graded and returned to you. Answers may or may not be posted on the web, depending on demand.

Problems

1. Use Farey series to approximate e, which is about 2.718281828. (Rather, use Farey series to approximate e-2, which is about 0.718281828.) What is the error in this approximation?
2. Test each of the following numbers for divisibilty by: 2^k, 3^k, 5^k, 7, 11, and 101:
   1. 741,954,888
   2. 1,039,715,607
   3. 44,774,865,619
3. The number 101,617 is the product of two ``large'' primes. Use Fermat's method to compute these two primes. (Hint: The square root of 101,617 is about 318.8.
4. 1. One of the conditions for divisibilty relies on the fact that
      

      10 x + y = 0 mod 7        if and only if        x - 2 y = 0 mod 7

      Prove this fact.

   2. In class I erroneously claimed that
      

      10 x + y = x - 2y mod 7

      for any x and y. Prove, by example, that this is not true.

   3. For what values of a does
      

      10 x + y = a mod 7        if and only if        x - 2 y = a mod 7

      hold for all x and y?

5. [Anton's Question] Recall that the condition for divisibility by 11 is that 11 divides the alternating sum of the digits. Anton asks: Is the alternating sum of the digits of 11^k always zero?

These problems are also available as a pdf file.

Solutions

Please email Peter if you are interested in answers or solutions for the web. Thanks.