## Problems from the Oct. 30th H.S. Math Circle

October 31, 2010You can discuss the answers to these problems here.

1.) Determine the length of the periodic part of the expansion of each number in the given base. Then determine the digits of the expansion itself.

a.) I can’t remember the first problem that I gave at the end of class. If someone remembers, please post it.

b.)

c.)

d.)

2.) What does it mean for a number to be rational in base b? Is it true or false that if a number is rational in base 10 then it will be rational in any integer base b > 1?

3.) Prove that between any two distinct rational numbers there exists an infinite number of distinct rational numbers.

4.) Convert each fraction in the given base to a reduced fraction in base 10.

a.)

b.)

c.)

5.) Find, as efficiently as possible, the value of:

a.)

b.)

c.)

d.)

e.)

6. Find:

a.) the smallest n such that . What does this mean about the decimal expansion of in base 10?

b) the smallest n such that . What does this mean about the decimal expansion of in base 10?

c.) the smallest n such that What does this mean about the decimal expansion of in base 6?

If you want a program that can compute using large numbers of digits, try PARI/GP: http://pari.math.u-bordeaux.fr/. It’s free, quick to install, and lots of fun!

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