Shadow Seminar for Number Theory
Problem Set
These problems are related in many ways to the math you teach in middle and high
school. Do the problems and explain how they connect to the math you teach.
1. How can modular arithmetic help you figure out if 2346 is a perfect square?
2. How can modular arithmetic help you figure out if 99416 is a perfect square?
3. Explain how modular arithmetic helps you find out that x2 – 5y = 27 has no integer
solutions.
4. For what integer values of n does n3 = 9k + 7? How does modular arithmetic help
find the values?
5. For what values of n does 75 = 35 mod n?
6. For what values of x does 35x = 14 mod 84?
7. Find the base five value of the base ten number 1416.
8. Find all the residues mod 56 that are 5 mod 8.
9. For what values of x does 91x = 26 mod 169? What simpler problem helps you solve
this equation?
10. Find the value of 1! + 2! + 3! + 4! + . . . + 99! for each of following:
a) mod 6
b) mod 12
c) mod 24
d) mod 48
e) mod 10
11. Show that 6 divides m3 – m for all integer values of m.
12. What is the remainder when 238 – 1 is divided by 31? What is the remainder when
21101 – 1 is divided by 31?
13. Find a closed formula for this function.
a1 = 3 and ak = 5ak – 1 + 2
14, Find a closed formula for this function and prove that it always holds.
(1 – 1/4) (1 – 1/9) (1 – 1/16) . . . (1 – 1/n2) =
15. For which values of n are the following true?
a) n2 > 2n + 1
b) 2n > n2
c) 2n > n!
16. For how many different positive integers n are each of the following true? Any
conjectures?
a) n = p2 – 9
CT 900 J3
b) n = p2 – 1
c) n = p2 – 16 d) n = p2 – 25
Carol Findell
Summer 2006