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

October 31, 2010

You 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.) $\left(\frac{1}{2}\right)_8$

c.) $\left(\frac{1}{5}\right)_7$

d.) $\left(\frac{1}{12}\right)_{23}$

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.) $\left(\frac{1}{3}\right)_5$

b.) $\left(\frac{5}{17}\right)_8$

c.) $\left(\frac{41}{50}\right)_6$

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

a.) $4^{31} \bmod 7$

b.) $17^{80} \bmod 12$

c.) $7^{57} \bmod 48$

d.) $5^{57} \bmod 48$

e.) $11^{5^{13}} \bmod 54$

6. Find:

a.) the smallest n such that $10^n \equiv 1 \bmod 29$. What does this mean about the decimal expansion of $\frac{1}{29}$ in base 10?

b) the smallest n such that $10^n \equiv 1 \bmod 31$. What does this mean about the decimal expansion of $\frac{1}{31}$ in base 10?

c.) the smallest n such that $6^n \equiv 1 \bmod 43$ What does this mean about the decimal expansion of $\frac{1}{43}$ 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!