MHB Show That $9$ Doesn't Divide $a^2-3$

  • Thread starter Thread starter tmt1
  • Start date Start date
AI Thread Summary
To show that 9 does not divide \(a^2 - 3\) for any integer \(a\), one can analyze the expression \(a^2 - 3\) modulo 9. By evaluating all possible values of \(a^2\) modulo 9, the results yield the set \(\{0, 1, 4, 7\}\) after subtracting 3. None of these results are congruent to 0 modulo 9, indicating that \(a^2 - 3\) cannot be divisible by 9. Therefore, it is concluded that \(9 \nmid (a^2 - 3)\) for any integer \(a\). This analysis confirms the original assertion.
tmt1
Messages
230
Reaction score
0
For any integer $a$, show that $$9 \nmid (a^2 -3) $$

I start with

$a^2 - 3 = 9r + k$, where $k \ne 0$ and $r$ is some integer but I'm unsure how to proceed
 
Mathematics news on Phys.org
Hi,
I must confess it took me longer to understand your starting point than to solve the problem.

[FONT=MathJax_Math]a[FONT=MathJax_Main]2[FONT=MathJax_Main]−[FONT=MathJax_Main]3[FONT=MathJax_Main]=[FONT=MathJax_Main]9[FONT=MathJax_Math]r[FONT=MathJax_Main]+[FONT=MathJax_Math]k, where [FONT=MathJax_Math]k[FONT=MathJax_Main]≠[FONT=MathJax_Main]0 and [FONT=MathJax_Math]r is some integer

I assume what you meant was:
For any integer $a$, if $a^2-3=9r+k$ for some integers $r$ and $k$, then 9 does not divide $r$. Basically, then you're talking about congruence mod 9.

For any integer $x$, 9 divides $x$ iff $x\equiv 0\pmod{9}$

So now compute all possible values mod 9 of $a^2-3$. You should quickly get the set $$\{0-3\pmod{9},\,1-3\pmod{9},\,4-3\pmod{9},\,7-3\pmod{9}\}$$

Quickly finish from here.
 
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...

Similar threads

Replies
7
Views
2K
Replies
1
Views
2K
Replies
1
Views
626
Replies
2
Views
1K
Replies
8
Views
2K
Replies
9
Views
3K
Replies
14
Views
2K
Back
Top