# integers

1. ### Order of integers and primitive roots

Let a,m be in Z with m>0. If a' is the inverse of a modulo m, prove that the order of a modulo m is equal to the order of a' modulo m. Deduce that if r is a primitive root modulo m, then r' is a primitive root modulo m.
2. ### Probably a simple explanation

Could someone explain why 2x-10 is not an irreducible in Z[x] but is an irreducible in Q[x].
3. ### consecutive odd integers

The sum of 3 consecutive odd integers is k. In terms of k, what is the sum of the 2 smaller of these integers? Do I need to use substitution somehow?? (The answer given is \frac {2k}{3} - 2 but can't figure it out..)
4. ### Primitive Roots - product of integers <= n and rel prime to n

(1) Show that if n is an integer which has primitive roots, then the product of the integers less than or equal to n and relatively prime to n == -1(mod n). (2) Show that (1) is not true if n does not have a primitive root.
5. ### Matrix of integers whose inverse is full of integers

I'm trying to come up with examples for my students to do in preparation for an upcoming test. We're not allowing them to use calculators, so the examples need to be relatively easy to compute by hand. In particular, I'm trying to come up with an algorithm to generate a matrix of integers whose...
6. ### integers

Determine whether there exist such integers x and y for which the following equation is satisfied: x^3+y^3-4=4xy(x+y) Thanks for any kind of input.
7. ### Integers sequence

Show that the sequence of positive integers given by a_{2n} = 3a_n -1, \ \ a_{2n+1}=3a_n +1 is strictly increasing. I have written down a reductio-ad-absurdum-proof, but it's based largely on tedious reduction of indices in a and eventually showing that the argument is proved by the...
8. ### eisenstein integers

could someone kindly show me how Eisenstein integers are Euclidean domain. I know that Eisenstein integers forms a Euclidean domain whose norm N is given by Z[ω] = ring of Eisenstein integers f(a + bω) = a^2 - ab + b^2 Thanks (Bow)
9. ### Integration and positive integers..

Using integration by parts, show that for any positive integer "n", integral (x^n)(e^x)dx = (x^n)(e^x) - n (integral) x^n-1(e^x) dx. Use the formula to determine integral (x^4)(e^x)dx I'm not really sure on this..but would it be... integral (x^n)(e^x)dx = (x^n)(e^x) - n (x^n-1/n-1)...
10. ### integers

How many integers between 1 and 101 are multiples of either 3 or 5 but multiples of both? A 20 B 33 C 45 D 47 E 53
11. ### Prove that all n>0 can be defined as sums of distinct powers of 2 via induction

Prove by induction that every positive integer n can be defined as a sum of distinct powers of 2, i.e. as a sum of the subset of the integers 2^0, 2^1, 2^2, and so on. (For the inductive step, consider the case where (k + 1) is even and the case where it is odd.) I found a variant online on...
12. ### set of odd integers proof

I am working on a simple set theory proof involving the definition of odd numbers, and so far I've done one containment. I would guess that if thiss is correct, then the other containment would be equally simple. Does this look alright so far? \mbox{If }A=\{x \in \mathbb{Z}~|~x = 2k+1\mbox{...
13. ### Nonnegative integers problem

Let N=30030, which is the product of the first six primes. How many nonnegative integers x less than N have the property that N divides x^3-1?
14. ### integers proof

Let x be an indeterminate. Let summation(i = 0 to n)a(subscript)i * x^i be a monic polynomial with integer coefficients and positive degree n. Assume that this polynomial has a rational root r. Prove that r is an integer.
15. ### Find any postive integers x,y,z,u

Find any postive integers x,y,z,u such that: x^{2}+y^{2}=z^{4} x+y=u^{2}
16. ### existence of integers

Suppose a,b are two integers with \gcd(m,n)=1. Prove that there exists integers m,n such that a^{m}+b^{n} \equiv 1 \mod{ab}
17. ### Prove that all integers are of the form...

Prove that all integers are of the form 3k, 3k+1, or 3k+2 The problem is clearly easy and it's obvious why this is true, but I just don't understand how exactly to go about proving it...help? Thanks!
18. ### How many six-digit positive integers are there, which is evenly divisible by 4 or 9?

How many six-digit positive integers are there, which is evenly divisible by 4 or 9? (If you let A be the amount of six-digit positive integers that are evenly divisible by 4 and B be the amount of six-digit positive integers that are evenly divisible by 9, so you should therefore determine the...
19. ### integers

Three integers are randomly selected without replacement from the set { 1,2,3,5,6,7}. What is the probability that the mean of the values chosen is less than, but not equal to, 5?
20. ### tricky question, prove for all positive integers n

Prove that for all x > 0 and all positive integers n e^x > 1 + x + x^2/2! + x^3/3!+ ... + x^n/n! Hint: e^x = 1 + \int _0\,^x\!e^tdt > 1 + \int _0\,^x\!dt = 1+ x e^x = 1 + \int _0\,^x\!e^tdt > 1 + \int _0\,^x\!(1+t)dt = 1+ x + x^2/2, and so on. Prove by induction on n...