# integers

1. ### how to prove multiplicative integers (mod p) has at least one generator?

How do you prove the multiplicative integers (mod p) has at least one generator? I know how to show a^(p-1) is congruent to 1 using Fermat's Little Theorem but how do you show p-1 is the smallest such integer? Thanks!
2. ### Homework help!!

Hello! I am looking for some assistance with a number theory homework assignment. I do NOT want it done for me though! I missed the first day of class so I am a bit lost on it. I am not even sure where to start or what theorems to use. I just need some direction and instruction on how to do...
3. ### Find all positive integers m and n, where n is odd, that satisfy 1/m+4/n=1/12

Find all positive integers m and n, where n is odd, that satisfy 1/m+4/n=1/12 
4. ### Confidence Intervals for not integers numbers ratio

Hi, I’m having a problem with a particular case of binomial proportion. I want calculate a confidence Intervals for a binomial proportion for an efficiency. This kind of intervals are usually defined for ratios between integers numbers but in my case I had to subtract from both numerators...
5. ### Confidence Intervals for not integers numbers ratio

Hi, I’m having a problem with a particular case of binomial proportion. I want calculate a confidence Intervals for a binomial proportion for an efficiency. This kind of intervals are usually defined for ratios between integers numbers but in my case I had to subtract from both numerators and...
6. ### Even integers problem

If the sum of the even integers between 1 and k inclusive is equal to 2k, what is the value of k? How do you prove that k must be 6 and not larger or smaller than 6?
7. ### Consecutive Integers

Find three consecutive odd integers that the sum of the first and second is 27 less than three times the third. The second question I have is the sum of an even number of consecutive odd integers is an _____ Integer.
8. ### Find the number of pairs (x,y) of positive integers which satisfy the equation x+2y=n

Let n be an odd integer >=5. 1) Find the number of pairs (x,y) of positive integers which satisfy the equation x+2y=n. 2) Find the number of triples (x,y,z) of positive integers which satisfy the equation x+y+2z=n. Anyone can help me? Thanks in advance.
9. ### Of 2005 integers whose product is even, at most how many can be odd?

How would you solve this? The question doesn't specify which integers, but my mentor tells me there is a solution. Could someone please reasonably explain this to me? Thanks!
10. ### Adding/Subtracting positive and negative integers. HELP!

Is there some kind of trick to help me understand what to do when adding or subtracting integers? I just don't understand how you do this and it's making me upset. As soon as I think I understand it I will get a problem, answer the way I think is right and it turns out to be wrong. I don't...
11. ### Prove that if m, d, and k are integers and d > 0, then (m + dk) mod d = m mod d.

Im having trouble with this proof. Could someone help me? Prove that if m, d, and k are integers and d > 0, then (m + dk) mod d = m mod d.
12. ### "Integers" in quadratic field

Greetings, Is it true that the subring of quadratic field, given by \{a+b\sqrt{D}|a,b \in Z\} is contained in the subring \{a+b\frac{1+\sqrt{D}}{2}|a,b \in Z\}, if D is equivalent to 1 modulo 4. Dummit and Foote use the terminology "slightly larger one" for the second, which implies that it...
13. ### If (x - a)(x - b)^3 has integer coefficients, then a, b are integers

Show taht if polynomial $(x -a) (x-b)^3$ has integer coefficients, then $a,b$ are integers.
14. ### Prove that for all integers n, n^2 - n + 3 is odd

I tried to think of a good way to prove this, would by induction be the best way this is not my formal write up yet for n=1 1^2 - 1 +3 = 3 so it is clearly true for n=1 suppose it is true for n = k; k \in \mathbb{Z} then k^2 - k + 3 = 2t+1 ; t \in \mathbb{Z} 2t+1 is an odd number number by...
15. ### for all integers a,b,and c if a|bc then a|b or a|c

for all integers a,b,and c if a|bc then a|b or a|c let a,b, and c be integers such that a|bc then, ak=b or ak=c (is this using what I need to prove to prove it??) this doesn't seem right
16. ### prove or disprove: The difference of the squares of any two consecutive integers is

prove or disprove: The difference of the squares of any two consecutive integers is an odd integer Let m, n and p be any integers such that m and n are consecutive and Let m^2 - n^2 = p. if m > n, then by definition m=n+1. Because m and n are consecutive integers neither are both even nor...
17. ### some integers

The solution to (2^x)(e^(3x+1))=10 is (p + ln q)/(r+ln s), where all of p, q, r and s are integers. D=p+q+r+s.
18. ### Representation of p-adic integers using primitive roots

Let p be an odd prime and let r be any positive integer that is a primitive root module p^2. Let X_p = \varprojlim \Bbb{Z} / (p-1)p^n \Bbb{Z} be the inverse limit of the rings \Bbb{Z} / (p-1)p^n \Bbb{Z}. This is useful because for any unit u \in \Bbb{Z}/p^n\Bbb{Z}, there exists K \in \Bbb{Z} /...
19. ### Number of k-subsets of an n-set, no pair of consecutive integers

Let f(n,k) be the number of k-subsets of an n set {1,2,....,n} that does not contain a pair of consecutive integers. Show that f(n,k)= {n-k+1} \choose {k} I know we can think of it as a sequence of k 1's and n-k 0's with no pair of consecutive 1's, but I can't figure out why this would be the...
20. ### Prove that out of two consecutive integers, one is divisible by 2

Hi. I'm supposed to prove that given any two consecutive integers a and a+1, one of them is divisible by 2. And I'm supposed to use the Division Algorithm. I'm not entirely sure where to begin. Any help is appreciated!