0
votes
1answer
36 views

Finding greatest common divisor between two polynomials.

I have the following past exam question: Calculate $\operatorname{gcd}(x^3 + 2x^2 + 2,2x^2 + 1)$ in $\mathbb{F}_3$ Now I haven't encountered this sort of gcd before(usually I am trying to solve ...
0
votes
0answers
18 views

WordProblem on factors and remainder theorem

Mr.Chaalu while travelling by Ferry queen has travelled the distance one Kilometer more, than the fare he paid per km. Initially he had total amount of Rs.350/- in his wallet. Now he is only left with ...
-2
votes
3answers
138 views

An intutive way to think about odd and even numbers. [closed]

What is an intuitive way to think about odd and even numbers? And about divisibility also...
5
votes
3answers
165 views

Prove $x^3-3x+4$ is irreducible in $\mathbb{Q}[x]$

I want to prove $x^3-3x+4$ is irreducible in $\mathbb{Q}[x]$. Eisenstein's criterion doesn't apply here, so I think the simplest method would be to use the Rational Roots Test, right? If I can use ...
0
votes
2answers
114 views

Bezout's Identity for polynomials

Im working out a problem where I find out the GCD of two polynomials using Euclid's Algorithm, and then I need to use Bezout's Identity to make $\gcd(r,s)=ra+sb$ The question gives me $x^5+1$ and ...
1
vote
1answer
108 views

Sums and differences of distinct factors

Given $k, n \in \mathbb{N}$, let $\tau_{k}(n)$ denote the $k$th positive factor of $n$ in strictly increasing order. For example, $\tau_{1}(6) = 1; \tau_{2}(6) = 2; \tau_{3}(6) = 3; \tau_{4}(6) = 6$. ...
1
vote
2answers
56 views

If I remove the premise $a\neq b$ in this question, will the statement still be true?

I encountered this proving problem, I can do the proof but my question is why in the condition/premise we need $a$ to be unequal to $b$? My guess is that even $a=b$, the statement is still true, is it ...
1
vote
2answers
88 views

GCD of the already GCD

Say $a$ and $b$ are integers. $\gcd(a,b)$ is then $d$. Now if $a$ equals $dm$ for some integer $m$ and b equals $dn$ for some integer n, how come the gcd of this m and n is always 1?
-1
votes
1answer
61 views

Efficient way to find lowest divisor of an integer.

I have followed the given way to find the lowest divisor of an integer, Let us assume n is the given integer. Check n is ...
1
vote
1answer
30 views

Computations question

a) Determine the prime factorizations of 3850 and 4125 b) Find the value of d = gcd(3850,4125) c) List all the positive divisors of d This is what I have so far. a) 3850: 11, 5, 5, 7, 2 4125: ...
1
vote
2answers
221 views

Rabin's test for polynomial irreducibility over $\mathbb{F}_2$

I know that $f(x) = x^{169}+x^2+1$ is a reducible polynomial over $\mathbb{F}_2$. I want to show this using Rabin's irreduciblity test. First, I then have to check if $f$ is a divisor of ...
0
votes
1answer
59 views

finding factors for gcd

To compute $gcd(25, 11)$, Euclid's algorithm would proceed as follows: $$\underline{25} = 2 \cdot \underline{11}+3$$ $$\underline{11} = 3 \cdot \underline{3}+2$$ $$\underline{3} = 1 \cdot ...
4
votes
1answer
157 views

Solve $a^3 + b^3 + c^3 = 6abc$

Find solutions for $a^3 + b^3 + c^3 = 6abc$ in $\mathbb{N}$, such that $gcd(a,b,c) = 1$, except for $(1,2,3)$ and its permutations. Using trial and error I found out that if $a,b,c$ are solution ...
6
votes
3answers
10k views

Find the largest prime factor

I just "solved" the third Project Euler problem: The prime factors of 13195 are 5, 7, 13 and 29. What is the largest prime factor of the number 600851475143 ? With this on Mathematica: ...
2
votes
2answers
296 views

Determine the number of factors for extremely large numbers.

An offshoot from a related question, is there a way to determine the number of possible factors (odd, even, prime, etc.) for extremely large integers without actually factoring them? Even an ...
2
votes
3answers
347 views

Factoring extremely large integers.

The question is about factoring extremely large integers but you can have a look at this question to see the context if it helps. Please note that I am not very familiar with mathematical notation so ...
6
votes
1answer
504 views

Smallest number with a given number of factors

From my rather rudimentary explorations of this fascinating problem, I believe it to be a layered and rewarding subject for investigation. My question, essentially, is: How do you find the smallest ...
1
vote
1answer
155 views

Interesting prime factorization function divisibility problem [duplicate]

Possible Duplicate: Is the set of all numbers which divide a specific function of their prime factors, infinite? Let the function $f(n) =(p_1^{a+1}-1)(p_2^{b+1}-1)... $ where $n$ is an ...