Math 350 - Fall 2002 - Number Theory

Homework #1, due Wednesday Sep. 18 , 2000.

1. (Schumer 1.2 # 24) Show that n^3/3 + n^2/2 + n/6 is an integer for n > 0

2. (Schumer 1.3 # 6)

  • a) What is the remainder when 100^100 is divided by 11 ?
  • b) What is the remainder when 702^10 is divided by 7 ?

    3. Prove that if p is a prime number p > 3, then p^2-1 is divisible by 24.

    4. (Schumer 1.4 #7) Prove that no matter which 1001 distinct integers are chosen from { 1, 2, 3, ..., 1991 }, two must have difference 9.

    5. Prove by induction that for every n, the binomial coefficient
    /  2n-1  \
    \ k /
    is odd, for all k with 0 <= k <= 2n-1.