NTIC Exercises

Find all simultaneous integer solutions to the following system of equations. (Hint: do what you would ordinarily do in high school algebra or linear algebra! Then finish the solution as we have done.)

$\begin{alignat*}{3} x \amp + y \amp + \amp z \amp = \amp 100\\ x \amp + 8y \amp + \amp 50z \amp = \amp 156 \end{alignat*}$

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11.

Compute the number of positive solutions to the linear Diophantine equation $6x+9y=c$ for various values of $c$ and compare to the three-case analysis at the end of Subsection 3.3.2.

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12.

Explore the patterns in the positive integer solutions to $ax+by=c$ situation in Section 3.3. For sure I want you to do this for the ones I mention there, but try some other values of $c$ and see if you see any broader patterns!

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13.

Prove that any line $ax+by=c$ which hits the integer lattice but $\gcd(a,b)\neq 1$ is the same as a line $a'x+b'y=c'$ for which $\gcd(a',b')=1\text{,}$ and explain why that means that without loss of generality Theorem 3.1.2 doesn't need any more explanations.

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14.

Find a primitive Pythagorean triple with at least three digits for each side.

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15.

Use Proposition 3.4.9 to prove that a Pythagorean triple triangle cannot have odd area.

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16.

Prove that 360 cannot be the area of a primitive Pythagorean triple triangle.

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17.

Find a way to prove that $x^4+y^4=z^4$ is not possible for any three positive integers $x,y,z\text{.}$ (Hint: use Corollary 3.4.13; this exercise needs a little cleverness.)

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18.

We already saw that if $x,y,z$ is a primitive Pythagorean triple, then exactly one of $x,y$ is even (divisible by 2). Assume that it's $y\text{,}$ and then prove that $y$ is divisible by 4.

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19.

Under the same assumptions as in the previous problem, prove that exactly one of $x,y,z$ is divisible by 3. (Combined with the previous exercise, this proves that every area of a Pythagorean triple triangle is divisible by 6. Is it also true that exactly one of $x,y,z$ is divisible by 5?)

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20.

A Pythagorean triple satisfies $x^2+y^2=z^2\text{.}$ Explore patterns for triples of positive integers which satisfy $x^2-xy+y^2=z^2\text{.}$ If Pythagorean triples correspond to right triangles, what sort of triangles do these triples correspond to?

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21.

Find a (fairly) obvious solution to the equation $m^n=n^m$ for $m\neq n\text{.}$ Are there other such solutions?

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22.

Show that

$\begin{equation*} \gcd(x,y)^2=\gcd(x^2,xy,y^2) \end{equation*}$

which we use in Proposition 3.7.2. You can try this using the set of divisors definition of gcd, or using the definition $\gcd(a,b,c)=\gcd(\gcd(a,b),c)\text{.}$

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23.

Explore Bresenham's algorithm in print or online. What is the connection to this chapter? How do non-solutions to linear Diophantine equations relate to actual solutions, in this context?

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24.

Assume you have relatively prime integers $a,b>0$ and a positive integer $k\text{.}$ Describe all $k-1$ positive solutions to $ax+by=kab\text{,}$ and use Definition 2.4.1 to find $k$ (positive) solutions to $ax+by=kab-1\text{.}$