1. S

    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. E

    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. D

    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. M

    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. O

    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. P

    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. M

    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. J

    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. A

    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. M

    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. P

    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. Y

    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. H

    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. G

    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. P

    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. Bernhard

    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. Bernhard

    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. R

    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. R

    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. Bernhard

    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...