Ian Morrison - Cleveland State University
Ian Morrison
MTH 493
Homework 1
1. (1) Find the solution set (if it exists), and compute one pair of integers that satisfies the equation [pic].
First, solve the equation [pic].
The gcd(234,321) = 3 from GCD algorithm in python. So let[pic].
Use the XGCD algorithm in python to find [pic]and[pic]. So [pic] and [pic].
18 is divisible by[pic], so [pic] has infinitely many solutions.
Let [pic].
To find one solution, solve the equation [pic], where [pic]and [pic].
The solution to the equation [pic] is [pic] and [pic].
Plugging [pic] and[pic] into [pic] yields [pic], which is true.
1.(2) Find the solution set (if it exists), and compute one pair of integers that satisfies the equation [pic].
The gcd(25,49) = 1.
[pic] and [pic] is obvious by inspection.
The solution to the equation [pic] is [pic] and [pic].
1.(3) Find the solution set (if it exists), and compute one pair of integers that satisfies the equation [pic]
First, solve the equation [pic].
The gcd(385,84) = 7 from GCD algorithm in python. So let[pic].
10 is not divisible by 7, so the equation has no integer solutions.
2. (1) Find [pic], if it exists. [pic].
The inverse exists because 8 is relatively prime to 31.
To find the inverse, solve the equation [pic].
Using XGCD algorithm yields [pic].
So [pic].
Check answer: [pic].
2.(2) Find [pic], if it exists. [pic].
The inverse does not exist because gcd(18,30) = 6.
2.(3) Find [pic], if it exists. [pic].
The inverse exists because 2 is relatively prime to 51.
To find the inverse, solve the equation [pic].
Using XGCD algorithm yields [pic].
So [pic].
Check answer: [pic].
[pic]
3.(1) Find all solutions of [pic]
Looking at the diagonal rows of the table shows that the solutions are [pic] and [pic].
3.(2) Find all solutions of [pic]
Looking at the diagonal rows of the table shows that there are no solutions.
3.(3) Find all solutions of [pic].
This is equivalent to [pic], which is congruent to [pic].
First inspect the diagonal row of the squares which contains only [0],[1],[4], and [7].
The only possible candidates for solutions are {[1],[4],[7]}, because [pic]is obviously not a solution.
[pic]is an obvious solution because [pic].
[pic]is a solution because [pic]by the table and [pic].
Likewise, [pic]is a solution because [pic]by the table and [pic].
So the solution set to the equation [pic] is [pic],[pic], and[pic].
4.(1) Calculate [pic]
Use the formula[pic]where[pic]are the prime factors of [pic].
The prime factors of 12 are 2 and 3.
So [pic]=[pic]
4.(2) Calculate [pic]
The prime factors of 144 are 2 and 3.
So [pic]=[pic]
5. Use Euler’s theorem to compute [pic].
According to Euler’s theorem, [pic] if [pic]
Gcd(5,847) = 1 obvious because 5 is prime and 847 not a multiple of 5.
In this case, [pic] and [pic](obtained by trial factorization and [pic] formula)
[pic].
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