Homework – Module 4 – 3303
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This is a 55 point assignment.
Homework rules:
Front side only. Keep the questions and your answers in order.
If you send it pdf, send it in a single scanned file. (dog@uh.edu)
If you turn it in personally, have the receptionist date stamp it and put it in my mailbox.
(651 PGH – 8am to 5pm)
Question 1
6 points
List the conditions and the formulas for the (x, y, z) of a Pythagorean Triple and the
radius of the inscribed circle. What mnemonic device would you use to remember these
formulas?
Give 3 additional primitive solutions to the ones from class.
What is the radius of the inscribed circle to each of these?
Sketch the 3 triangles with the inscribed circles, in cm, to scale.
How will you use this information in your teaching?
Question 2
6 points
Recall the Euclidean Algorithm way to find the gcd of two natural numbers.
Describe in a teaching style. How would you teach it to a 5th grader?
Find two very different additional ways to find the gcd. Describe these in a teaching
style.
Show examples of each style with the numbers 36 and 210. What’s the gcd and how do
you find it – three different ways.
Question 3
5 points
Theorem 1
LDE are solvable iff the gcd of the coefficients of x and y also divides the
constant.
Theorem 2
If ax + by = c has solutions, the number of steps to the Euclidean
Algorithm to find a linear combination of a and b that equals the gcd is
less than or equal to 5 times the minimum of {a, b}.
Theorem 3
If you have a Diophantine Solution ( x 0, y 0 ), you can get a whole family of
solutions to ax + by = c from
x = x0 
b
t
gcd
y = y0 
a
t
gcd
for any integer t.
Find 3 different point pair Diophantine solutions to:
Graph your answer.
5x + 2y = 100
Use all 3 theorems CLEARLY in your excruciatingly detailed work.
Question 4
5 points
Using the method from class (carefully adjusting to work with the negative 15), find a
family of Diophantine Solutions to
20x 15y = 30
Question 5
6 points
For the example on page 71:
explain how it is that the points X3, X4, and X5 are integral distances from X1 and X0
(the origin) and the point (0, 10395). Use a diagram and make it clear to the most
pedestrian of observers that you know what you’re talking about.
Question 6
6 points
Use the formula below to come up with two Diophantine quadruples.
{n, n+2, 4n+4, 4(n+1)(2n+1)(2n+3)} with the proviso that n is a natural number.
In a Diophantine quadruple, the following theorem is true:
If you multiply any two distinct elements of the set and add 1, the result is a perfect
square.
Show 2 examples, one from each of your triple, that illustrate the theorem.
Question 7
6 points
Brilliant numbers are numbers with prime factors all of the same length.
For example:
22 is not brilliant because its prime factors (2 and 11) are of different lengths.
However, 35 is brilliant because its prime factors (3 and 5) are of the same length.
Brilliant numbers of with n factors of the same length called n-brilliant.
253 is a 3 digit, 2-brilliant number. It’s prime factors are 11 and 23.
What is the minimum 2-brilliant number? List as many number name descriptors of this
number as you can…ie is it abundant, composite, part of a Pythagorean triple, evil, a
palindrome…
The minimum 3-brilliant number? List as many number name descriptors of this number
as you can.
Give a 3 digit 2-brilliant number other than 253. What are all the facts you can find out
about this number?
Question 8
8 points
Here’s a theorem about Pythagorean Triples. Give four primitive examples that illustrate
this theorem:
If (x, y, z) is a Pythagorean triple where z is the hypotenuse, then one of x or y is a
multiple of 3 and one of the numbers in the triple is a multiple of 5.
Question 9
2 points
Divide 100 into two summands such that one is divisible by 7 and the other by 11.
This is a problem posed by Euler in 1770. Set it up and solve it using the method taught
in class for Diophantine equations.
Question 10
5 points
Rewrite the proof on page 68 that the radius of the incircle of a Pythagorean triangle is
n(m  n) in your own words. Keep it to one side of one sheet. Make sure I can tell you
know what you’re talking about.