Integer Solutions
-For lower values of c, all integer solutions have the property that either x or y or both are
negative. However for higher c, there are integer solutions where both x and y are nonnegative. For example, 4x + 9y = 60 has the solution (x
Diophantine Examples
-Find the non-negative integer solutions of 10x + 15y = 100.
We know that the full set of integer solutions is:
cfw_(x = 20 + 3k, y = 20 2k) | k Z
We need to restrict ourselves to k such that
20 + 3k 0
20 2k 0
These are equivalent to:
Greatest Common Divisor
-Imagine you have a floor of size 1053 inches by 481 inches. You would like to tile
your floor with square tiles. You can of course use 1 inch by 1 inch tiles, but you would
like to use as few square tiles as possible. No partial t
Bezout's Identity
-Let a, b, c Z, where d = gcd(a, b) and c
is a multiple of d. Suppose that (x = x0, y = y0) is one particular integer solution to
ax + by = c.
-Then the complete set of integer solutions is
S = cfw_x = x0 + k b/d, y = y0 k a/d
kZ
Proof:
Division Theorem
-If a, b Z, with a 6= 0, we say that a divides b, written a|b, if q Z, such
that b = aq. We say that a is a divisor or factor of b.
-Let a, b, c Z, where a 6= 0 and b 6= 0. Then:
-If a|b and b|c, then a|c.
-If a|b and a|c, then a|(bx + cy
Diophantine Equations
-Imagine that your favorite restaurant is McDonalds, and your favorite food is chicken
nuggets.
It turns out that chicken nuggets come in two different sizes: the 4-box and the 9-box.
Youre
having your friends for dinner. Fortunately
Sylvester's Theorem
-Let a and b be positive, relatively prime
integers and c be a positive integer. Then there exist non-negative integers, x and y, that
satisfy
ax + by = c,
if
c > ab a b .
When c = ab a b, there are no non-negative integer solutions.
P
Euclidean Algorithm
-Suppose we want to find integers x and y such that 89x + 33y = 1.
To find x and y, we will need to first apply the Euclidean algorithm. This seems
extraneous,
since we already know that gcd(89, 33) = 1. However well need to do this, s
Properties of Euclidean Algorithm
-To solve the Chicken McNugget problem, it will help to first realize that there are actually
an infinite number of different pairs, (x, y), of integer solutions to an equation of the form
ax + by = c,
when a and b are re
Bezout
-Question: Do you see any integer solution for 10x + 15y = 1?
Question: How about 10x + 15y = 2? Whats the problem?
Answer: Seems like the only hope is if c is some multiple of 5.
Question: Whats special about 5?
Answer: 5 = gcd(10, 15).
-Let a, b,