# integers

1. ### Prove that integers are closed under addition

Prove that the sum of two integers is also an integer. This is from an advanced calculus book. These are the facts presented earlier in the book which I assume are supposed to be sufficient to prove this: \mathbb{Z}=\text{the natural numbers, their negations, and zero.} \text{If...
2. ### Prove by interpreting the parts in terms of compositions of integers. Combinatorics.

Given the identity \sum_{i=0}^{n} \binom{k-1+i}{k-1} = \binom{n+k}{k} Need to give a combinatorial proof by interpreting the parts in terms of compositions of integers (neither by induction nor using subsets) :) Please, help!
3. ### Does There Exists Three Consecutive Odd Positive Integers That Are Primes

Does There Exist Three Consecutive Odd Positive Integers That Are Primes Hello, The exact question is, Prove or disprove that there are three consecutive odd positive integers that are primes, that is, odd primes of the form p,p+2, and p+4. At first glance, I did not realize that this...
4. ### Suppose that m,n,q,r are integers satisfying n=mq+r. Then gcd(m,n)=gcd(m,r).

Suppose that m,n,q,r are integers satisfying the identity n=mq+r. Then gcd(m,n)=gcd(m,r). Let gcd(m,n)=k where k is some integer, then k|m and k|n. n=mq+r can be expressed as r=n-mq=k|n+k|m. Therefore k|r. Let gcd(m,r)=p where p is some integer, then p|m and p|r. n=mq+r=p|m+p|r. Therefore...
5. ### integers role in mathematics

why do integers play such a vital role in mathematics?
6. ### Ring based on families of integers - The Function Ring

I am trying to get a full understanding of Cohn Example 4 on page 9 - see attachment. A formal and rigorous example showing clearly the nature and role of the families (and their indicies) involved would help enormously. Cohn, page 9, Example 4 states: "Let I be any set and denote by...
7. ### show that a^2+b^2=3c^2 has no solutions in the integers

How I'm showing that the equation a^2+b^2=3c^2 has no solutions (other than the trivial solution (a,b,c)=(0,0,0)) in the integers? Thanking you in anticipation
8. ### Integers

How many pairs of positive integers x and y satisfy the equation xy=20132013?
9. ### Integers

How many pairs of positive integers x and y satisfy the equation xy=20132013?
10. ### a fun proof of inf integers, tell me what you think.

Thought of this today. two years ago someone told me that you cannot know if there exist a biggest integer or not. then i said that there must be a bigger integer, because we can add one to it. But now i thought \infty+1 = \infty, so when we get to \infty big numbers adding one does not...
11. ### Purpose of subtracting negetive integers

I'm an old man restudying per-algebra. Ok, not real old, but it's been many years since I studied math. What is the purpose of subtracting negetive integers? Or should I say, when or where would I need to do that? Practical application. Perhaps it's needed to understand algebra later on...
12. ### Possible Combinations Question. Please Assist.

Hello all, I have been hoping to solve a problem involving all possible combinations for a set of integers. I started scribbling combinations out on paper, but I thought there might be some type of equation that could tell me the maximum possible combinations given a certain scenario. In...
13. ### Integers

Set S consists of integers from 2-30, inclusive. The number of composite numbers in S is how much greater than the number of prime numbers in S? A. 6 B. 7 C. 8 D. 9
14. ### Integers

Hello, I am trying to solve this problem: If j and k are integers and j - k is even, which of the following must be even? a) k b) jk c) j + 2k d) jk + j e) jk - 2j According to the booklet the answer is d, however how would I even go about arriving at this answer? Any help would be...
15. ### ∀n(3n ≤ 4n) Domain: all integers ??

There is a problem in my Discrete Math book concerning Universal Quantification: ∀n(3n ≤ 4n) Domain: all integers True or False I answered false because of negative integers. The answer in the back of the book says it's true. Is that a mistake on the books part or am I missing something here?
16. ### number of integers bet 90 and 990

How many integers bet 90 and 990 are divisible by 7? is there any short and simple way of finding the answer? thanks
17. ### Not completely sure how to prove this

Please help with this: By considering the possible residue classes modulo 5 of a square, or otherwise, prove that if p^2 + q^2 + r^2 is divisible by 5 then 5 divides at least one of p,q,r
18. ### Have no idea on how to prove

Can someone help me prove that 2^3^n + 1 is divisible by 3^(n+1)??!!
19. ### Can't prove this equation

I need help on how to prove that 9^(n+3) + 4^n is divisible by 5. Please help, I have no idea of how to solve this
20. ### are inductive proofs really complete if they only hold for integers?

if inductive proofs are only proved for integer values of n then is there a way to complete the proof for all values of n?