# divisors

1. ### how can I find the number of common divisors

Hello, If I have two integers a and b and I know their gcd, then how can I find the number of common divisors between a and b? Thanks
2. ### What are the elementary divisors of U(Z/nZ) (group theory)?

Hi: The problem: What are the elementary divisors of U(n), the multiplicative group of all congruence classes [a] mod n with (a,n)=1? Notation: a = b means a equal b or a isomorphic to b depending on context. I find that U(p^m q^n) = U(p^m) x U(q^n), where p,q primes, using the Chinese...
3. ### Zero divisors of a polynomial factor ring

Problem As we've observed, a(x) = x^2 + x + 1 2 Z_3[x] is reducible. According to the theory (Theorem 17.5), Z3[x]=(a(x)) is not a field. Find a non-0 element q(x)+(a(x)) that has no inverse. Suggestion: Find a 0-divisor of Z3[x]=(a(x)). Attempt at a solution: I've factored x^2 + x + 1 into...
4. ### How many zero divisors in a ring

If I want to know how many zero divisors there are in the ring \mathbb{Z}_5[x]/(x^3-2), how do I go about solving that? I've noted that for p(x) = x^3-2, p(3) = 0 \space \space (\mathrm{mod} 5) and p(5) = 0 \space \space (\mathrm{mod} 5). Does that make 3 and 5 zero divisors? If I could write...
5. ### How do I find the divisors of a positive integer?

I've read many sites that try to explain how to find all the divisors of a positive integer but none of them seem to make sense. I know there's some sort of formula that can be used. I need to know this to answer one of my questions in my homework. Any help is greatly appreciated (Happy) The...
6. ### sum of k-powers and prime divisors.

Show that for every finite set of integers $\left\{ a_{1},\dots,a_{n}\right\} ,\, n>2$, the set of prime factors of the elements of the set: $$\left\{ \overset{n}{\underset{i=1}{\sum}}a_{i}^{k},\, k\in\mathbb{N}\right\}$$ is infinite.
7. ### Zero divisors and cancelation as technique for solving equations ...

I have the feeling that this question will be another manifestation of grand-scale stupidity, but anyways ... I'm trying to put it all together. In a ring R which has zero divisors, what and when can we cancel, when trying to solve an equation? Known facts: 1. An element is either zero or...
8. ### number of divisors

let p_{i} be a prime number and a_{i} be a positive integer for all i a) list all distinct divisors of 32 1,2,4,8,16,32 2^5 5+1 = 6 divisors b) how many distinct divisors does p^a have? a+1 divisors c) how many distinct divisors does p_{1}^{a_{1}}p_{2}^{a_{2}}\cdot\cdot\cdot...
9. ### Abstract Alegbra - Zero Divisors and Units

Z2 × Z4 is a ring under component-wise addition and multiplication: (a, b) + (c, d) = (a + c, b + d) and (a, b) · (c, d) = (ac, bd). Classify each element of Z2 × Z4 as a zero divisor, a unit, or neither. If the element is a zero divisor, ﬁnd a nonzero element whose product with the...
10. ### What is the smallest positive integer with exactly 768 divisors?

Hi everybody, What is the smallest positive integer with exactly 768 divisors? I think that 2^767 might have 768 divisors, but I think there would be a smaller one. I'm thinking that it would be some product of prime numbers. Like \:2^x\times3^y\times5^z\; etc Anyone got any ideas. Thanks.
11. ### Integers GMAT

If n-9p, where p is a prime number greater than 3, how many different positive divisors does n have? This is a question from the subject: integers. I have no idea how to solve this problem. I keep guessing, but whats the trick to this question? can you give me the actual steps to finding the...
12. ### Sum of Positive Divisors Function

Let n \in Z with n > 0 Prove that: (\sum_{d|n, d>0}\upsilon(d))^2 = \sum_{d|n, d>0}(\upsilon(d))^3 [Hint: It suffices to prove the equation above for powers of prime numbers (Why?)]
13. ### Greatest Common Divisors

Show that, for any positive integers a,b the set of all ma+nb (m,n positive integers) includes all multiples of (a,b) larger than ab. This problem is from a section on the integers in Ch 1 of Birkhoff and MacLane which covers: (a,b), gcd of a,b [a,b], lcm of a,b Division algorithm, a=bq+r...
14. ### Prime Divisors Positive Difference

For a composite positive integer x, denote by pd(x)the smallest positive difference between any two prime divisors of x. Find the smallest possible value of pd(x) for composite x of the form x=n^100+n^99+...+n+1, where n is a positive integer. It is a very typical question. after solving for a...
15. ### Finding a conjecture relating prime numbers to divisors

Hello I was just wondering if anyone could help me find a conjecture, theorem, hypothesis around this particular result. If p,q are prime numbers, m,n are integers, and f(r) is the number of nontrivial divisors of r (ie. not 1 or r itself). That if r=(p^m)*(q^n) (p≠q) that f(r)=(m+1)n+(m-1)...
16. ### Polynomial rings - zero divisors in R and r[x]

Show that if R has no zero divisors then R[x] also has no zero divisors.
17. ### Polynomial rings - zero divisors

Let R = Z/4Z Find all the zero divisors of R[x] ... ... and then, further, find all elements of R[x] that are neither units nor zero divisors.
18. ### Relations between the set consisting 0 and zero divisors and prime ideals.

One exercise from my textbook: The set consisting of 0 and all zero divisors in a commutative ring with identity contains at least one prime ideal. Having thought about this problem for days, I only come up with the solution to the simplest case in which there's no zero divisors in such a...
19. ### Natural divisors

Hi, I have recently encountered a problem I was unable to solve. I have tried everything and none of it worked. Please help. The problem goes like this: Prove that every natural odd number 'n' undividable by number 5, divides at least one of the numbers 1, 11, 111, 1111, 11111, 111111...
20. ### Existence of Greatest Common Divisors - Dummit and Foote Ch8

I am reading Dummit and Foote Ch 8 - Euclidean, Prinipal Ideal and Unique Factorization Domains. On page 274 D&F write: (see attached) "Thus a greatest common divisor (if such exists) ... ..." Even when two elements of a ring are prime a gcd exists ... ??? Can anyone give me examples of...