Problems

Problems: Fall 2004

• Proof: Prove that there are infinite primes.
• What is the (Tau function) Τ(1000000).
• What is the (Euler function) Φ(1000).
• If m = 2004! (factorial), how many trailing zero's does m have.
• Prove that if the sum of the digits of a number is divisible by 3 than the number is divisible by 3. Then show what are the conditions for divisibility by 11 and then prove it.
• Find the dimension of the Serspinsky Triangle.

Problems: Spring 2005

• If M and N are the coefficients such that (x^2)+ 1 is a factor of (5x^4) + (4x^3) + (3x^2) + Mx + N, find the product of M and N.

• Three different numbers are chosen such that when each of the numbers is added to the average of the other two, the results are 65, 69, and 69 respectively. Find the average of the original three numbers.

•  Let A be one of a regular hexagon whose side length is one meter. To the nearest centimeter, what is the sum of the distances from A to the other five vertices of the hexagon?