Number Theory Midterm
Study Guide
Review Session – Monday March 1 in IT 222
Some key definitions/Theorems to know:
a. Well Ordering Principle (We assumed this, we never proved it. It’s the assumption the
whole course comes from.)
b. First and Second Principles of Finite Induction
c. Pascal’s Rule
d. Binomial Coefficient
e. Binomial Theorem
f. Triangular Number
g. Division Algorithm
h. Greatest Common Divisor
i. Relatively Prime
j. Euclid’s Algorithm
In addition, you should know theorems 2.2-2.5 Note:
2.4 tells us a and b are relatively prime iff there exist integers x and y with ax+by=1.
2.5 says that if a|bc with a and b relatively prime, then a|c
You do not need to study 2.6 for the midterm.
You should be able to apply the Euclidean Algorithm to do both:
a. Given integers a and b, find their gcd
b. Given integers a and b, express the gcd of a and b as a linear combination.
You should also recall from page 18 that the square of any integer is of the form 4k or 4k+1
for some k and that the square of any odd integer is of the form 8j+1 for some j. (You
should also be able to prove these, but unless asked to prove, you can assume them. )
Some example problems: (Two of the test questions will come from these. I expect to add
one more problem on Monday to this list.)
1. Show that 2.6.10… (4n-2)=(2n)!/n!
2. More than any other method, we’ve used the well ordering principle to prove theorems.
Don’t complete the proof, but explain how we used the WOP to set up the proof that given 2
integers, not both 0, there exists x and y such that gcd(a,b)=ax+by (Theorem 2.3)
3. Show that the cube of any integer has one of the forms 9k, 9k+1 or 9k-1 for some k.
4. Expand (a+b)7
5. Find the gcd(1617,525)
6. Find x and y so the gcd(24,138)=24x+138y using the Euclidean Algorithm
7. For any integer x, show a|(ax+b) iff a|b
8. Show that if a|bc then a| gcd(a,b)gcd(a,c)
9. (Added Monday night)
The nth Catalan number for n  0, is defined to be
Prove
Cn 
22n 1
Cn1 for n1.
n 1


Cn 
1 2n 
 
n 1  n 