Prove Number Theory Proofs: Sum Irrational, (m+dk) mod d, x^2=x, n^2 mod 3

In summary: So for n=1 mod 3, n^2=1 mod 3 and for n=2 mod 3, n^2=4 mod 3 which is equivalent to 1 mod 3. Therefore, n^2 mod 3=1. In summary, for any integer n not divisible by 3, n^2 mod 3=1.
  • #1
and1bball4mk
4
0
1. For any positive integer n, if 7n+4 is even, then n is even.
2.Sum of any two positive irrational numbers is irrational.
3. If m, d, and k are nonnegative integers with d=/=0 then (m+dk) mod d = m mod
4. For all real x, if x^2=x and x=/=1 then x=0
5. If n is an integer not divisible by 3, then n^2 mod 3=1

2. Basically I have to prove (or disprove) all of those and I'm stuck. Any advice and feed back would be appreciated.
3. Here are my attempts at solutions
1. By contraposition, if n is odd then 7n+4 is odd. If N is odd, then n=2k+1 for some integer k. so 7(2k+1)+4, and by algebra we get 2(7k+5)+1. 7k+5 is an integer, so it must be odd.

Is this right? I think it is but I'm never sure because I'm terrible at number theory.

2. For this one I found a counterexample and I think that is sufficient to get the problem right but I want to know why it's a false statement..anyone have any insight?

3. This is confusing. Basically after half an hour of writing stuff I haven't reached any conclusions. I'm sure it has something to do with the quotient remainder theorem because the form of (m+dk) is very similar to QRT where given any interger n and positive integer d, there exists unique integers q and r such that n=dq+r. Any help would be appreciated.

4. This one is really bothering me. I know that it's true but I'm trouble proving it. I keep going back to the method of exhaustion but obviously that won't work. I have a feeling that this is really simple and I'm just overlooking something.

5. This is another that I know is true but I don't know how to prove it. I think part of the problem I'm having here is defining n as a integer not divisible by 3.Any help on any of those would be appreciated.

Thanks

 
Last edited:
Physics news on Phys.org
  • #2
Actually I might of made some headway on number 4.

If x^2=x then x=x/x. So x/x will always reduce to 1 unless x=0.

Is this sufficient for a proof?
 
  • #3
You're likely to get more responses if you put each problem in itw own thread, rather than posting a whole slew of them all at once.
 
  • #4
and1bball4mk said:
Actually I might of made some headway on number 4.

If x^2=x then x=x/x. So x/x will always reduce to 1 unless x=0.

Is this sufficient for a proof?

and1bball4mk said:
4. For all real x, if x^2=0 and x=/=1 then x=0
Which is it? x^2 = 0 or x^2 = x?
 
  • #5
Mark44 said:
Which is it? x^2 = 0 or x^2 = x?

It is x^2=x
 
  • #6
For 5, if n is an integer not divisible by 3, then its remainder when divided by 3 has to be either 1 or 2. I.e., n = 1 mod 3 or n = 2 mod 3.

Investigate each case to say something about n^2.
 
  • #7
Mark44 said:
For 5, if n is an integer not divisible by 3, then its remainder when divided by 3 has to be either 1 or 2. I.e., n = 1 mod 3 or n = 2 mod 3.

Investigate each case to say something about n^2.


Thanks.
 

1. How do you prove that the sum of two irrational numbers is irrational?

To prove that the sum of two irrational numbers is irrational, we use the proof by contradiction method. We assume that the sum of two irrational numbers, x and y, is rational. This means that x+y can be expressed as a ratio of two integers, a/b. From there, we can manipulate the equation to show that x and y must also be rational, which contradicts our initial assumption that they are irrational. Therefore, the sum of two irrational numbers must be irrational.

2. What is the significance of (m+dk) mod d in number theory proofs?

(m+dk) mod d is often used in number theory proofs to show that two numbers are congruent modulo d. This means that when dividing both numbers by d, the remainder is the same. This concept is important in many number theory proofs, including proofs involving prime numbers and divisibility.

3. How can you prove that x^2=x for all real numbers?

The equation x^2=x holds true for all real numbers when x=0 or x=1. To prove this, we can use the definition of equality, which states that two expressions are equal if and only if they have the same value for all possible inputs. By plugging in x=0 or x=1, we can see that both sides of the equation are equal, thus proving that x^2=x holds true for all real numbers.

4. What is the significance of n^2 mod 3 in number theory proofs?

n^2 mod 3 is often used in number theory proofs to determine the remainder when n^2 is divided by 3. This can help in proving certain properties of integers, such as whether they are even or odd. Additionally, n^2 mod 3 is also used in proofs involving quadratic residues and modular arithmetic.

5. Can you provide an example of a proof involving these concepts?

Sure, one example is proving that the product of two consecutive integers is always divisible by 2. We can start by stating that the two consecutive integers are n and n+1. Using the definition of congruence, we can write this as n(n+1) ≡ 0 (mod 2). Then, by using the distributive property, we get n^2+n ≡ 0 (mod 2). Since n^2 mod 2 = 0 and n mod 2 = 1, the equation simplifies to 1 ≡ 0 (mod 2), which is a contradiction. Therefore, the product of two consecutive integers is always divisible by 2.

Similar threads

Each integer n>11 can be written as the sum of two composite numbers?
    • Calculus and Beyond Homework Help
Replies
7
Views
2K
Prove by induction or otherwise, that ##\sum_{r=1}^n r(3r-1)=n^3+n^2##
    • Calculus and Beyond Homework Help
Replies
20
Views
1K
If ## p\neq5 ## is an odd prime, prove that either ## p^{2}-1 ## or....
    • Calculus and Beyond Homework Help
Replies
15
Views
2K
Solve the simultaneous congruence modulo equation
    • Calculus and Beyond Homework Help
Replies
3
Views
1K
If p##\geq##5 is a prime number, show that p^{2}+2 is composite?
    • Calculus and Beyond Homework Help
Replies
3
Views
737
Prove that if p > 3 is a prime number, then p^2 = [....]
    • Calculus and Beyond Homework Help
Replies
16
Views
2K
Prove ##a \cdot 1 = a = 1 \cdot a## for ##a \in \mathbb{N}##
    • Calculus and Beyond Homework Help
Replies
1
Views
512
Integral of x^n using Reimann sums
    • Calculus and Beyond Homework Help
Replies
4
Views
651
Prove ##1 + a=s(a)=a+1## for ##a \in \mathbb{N}##
    • Calculus and Beyond Homework Help
Replies
1
Views
574
Prove ##|\{ q\in\mathbb{Q}: q>0 \} |=|\mathbb{N}|##.
    • Calculus and Beyond Homework Help
Replies
3
Views
549

Hot Threads

Recent Insights

Back
Top