0

I have the answer for $4x\equiv 13$ (mod 47) but I can't quite understand it, could you please help?

Steps:

  1. $4x \equiv 13$ (mod 47)
  2. Noting that $4\times 12 = 48 \equiv 1$ (mod 47)
    I understand that $48\equiv 1$ (mod 47), but I don't understand where this comes from. Am I supposed to find a coefficient (in this case 12) that I can multiply the LHS leading to $ \equiv 1$? if so, why?
  3. leading to $48x \equiv 12 \times 13$ (mod 47)
    Ok, I understand I can multiply both sides by 12 with no problem
  4. That is, $x\equiv 3\times 4 \times 13 \equiv \ldots \equiv 15$ (mod 47)
    I can calculate the above, but what happened to the 48?!
  5. $x=15+47t$
    I can understand this, if the other steps are correct.

Thanks!

TheVoiceInMyHead
  • 187
  • 2
    (2) "...if so, why?" Because if we were to talk about arithmetic in a more normal scenario we would have said "Oh, well, $4x=13$... let us 'divide' both sides by $4$" Well, here we can't really "divide" but we can "multiply by multiplicative inverses" if such things exist. Here we have $4x\equiv 13$ and so $4^{-1}\times 4x \equiv 4^{-1}\times 13$ and the $4^{-1}\times 4$ cancel eachother out. – JMoravitz Oct 05 '22 at 15:25
  • 2
    (4) "what happened to the 48?!" It is still there. Remember that in this context of mod47 we have that $48$ and $1$ are the same "number" (at least while they are on the ground and not in an exponent). That is to say, if $A\equiv a$ we have $A+b\equiv a+b$ and we have that $A\times b \equiv a\times b$... we can replace numbers by other numbers which are equivalent and it not change the meaning. Here, we just replaced the $48$ with $1$... and since multiplication by $1$ doesn't do anything, we left it invisible. – JMoravitz Oct 05 '22 at 15:26
  • are you familiar at all with group theory? If so, there's a fairly simple explanation for why there is a solution to $4x \equiv 13 \pmod{47}$. – user1090793 Oct 05 '22 at 15:28
  • The $12$ can be computed in this way. The idea is that because $\gcd(4,47)=1$, then the Euclidean algorithm is going to produce $a,b$ such that $4a+47b=1$. Note that this equation implies that $4a\equiv 1\pmod{47}$. The $a=12$ comes from there. – plop Oct 05 '22 at 15:40
  • 2
    $\bmod 47!:\ \color{#c00}{13\equiv 60}\Rightarrow x\equiv \dfrac{\color{#c00}{13}}4\equiv \dfrac{\color{#c00}{60}}4\equiv 15.,$ See my answers in the linked dupes for this and many other quick methods. – Bill Dubuque Oct 05 '22 at 16:12

0 Answers0