MHB Is $\gcd(n, n+2) = 1$ or $2$ for any integer $n$?

AI Thread Summary
The discussion centers on the properties of the greatest common divisor (gcd) for various integer pairs. It concludes that for any integer n, gcd(n, n+1) is always 1, making statement (a) true. Statement (b) is deemed false, as gcd(n, n+2) can be either 1 or 2, depending on whether n is odd or even. Statement (c) is confirmed as true, since gcd(n, n+2) equals gcd(n, 2), which is 2 for even n and 1 for odd n. Finally, statement (d) is validated using the gcd identity, confirming that gcd(n, n^2+m) equals gcd(n,m) for all integers n and m.
Guest2
Messages
192
Reaction score
0
Decide which of the following is true or false. If false, provide a counterexample.

(a) For any integer $n$, $\gcd(n, n+1) = 1$.

(b) For any integer $n$, $\gcd(n, n+2) = 2$.

(c) For any integer $n$, $\gcd(n, n+2) = 1$ or $2$.

(d) For all integers $n, m:$ $\gcd(n, n^2+m) = \gcd(n,m)$.

I think (a) is true and (b) is false since $\gcd(3,5) = 1.$
 
Mathematics news on Phys.org
Guest said:
I think (a) is true and (b) is false since $\gcd(3,5) = 1.$
I agree.

Guest said:
(d) For all integers $n, m:$ $\gcd(n, n^2+m) = \gcd(n,m)$.
This can be answered using the identity $\gcd(a,b)=\gcd(a,b-a)$.
 
Evgeny.Makarov said:
I agree.

This can be answered using the identity $\gcd(a,b)=\gcd(a,b-a)$.
Thank you. Is it

$\gcd(n, n^2+m) = \gcd(n, n^2+m-n) =\gcd(n, n^2+m-n-n)= \cdots = \gcd(n, n^2+m-\ell)$ where $\displaystyle \ell = \sum_{k=1}^{n}n = n^2$?
 
Yes, your reasoning for (d) is correct.
 
Evgeny.Makarov said:
Yes, your reasoning for (d) is correct.
For (c) I get $\gcd(n, n+2) = \gcd(n, 2)$ which is $2$ if $n$ is even, or $1$ if $n$ is odd. So it's true, I think.
 
Guest said:
For (c) I get $\gcd(n, n+2) = \gcd(n, 2)$ which is $2$ if $n$ is even, or $1$ if $n$ is odd. So it's true, I think.
Yes.
 
Suppose ,instead of the usual x,y coordinate system with an I basis vector along the x -axis and a corresponding j basis vector along the y-axis we instead have a different pair of basis vectors ,call them e and f along their respective axes. I have seen that this is an important subject in maths My question is what physical applications does such a model apply to? I am asking here because I have devoted quite a lot of time in the past to understanding convectors and the dual...
Insights auto threads is broken atm, so I'm manually creating these for new Insight articles. In Dirac’s Principles of Quantum Mechanics published in 1930 he introduced a “convenient notation” he referred to as a “delta function” which he treated as a continuum analog to the discrete Kronecker delta. The Kronecker delta is simply the indexed components of the identity operator in matrix algebra Source: https://www.physicsforums.com/insights/what-exactly-is-diracs-delta-function/ by...
Back
Top