Understanding the answers of $x^2 \equiv 28 \pmod 6$

Question

Solving $x^2 \equiv 28 \pmod 6$

The answers are: $x \equiv 2 \pmod 6$ and $x \equiv 4 \pmod 6$.

When I plugged in to check the answers, it would be $2^2 \equiv 4 \pmod 6$ and $4^2 \equiv 16 \equiv 4 \pmod 6$. It is $4$ but not $28$, why is this, I don't understand? Anyone can help me understand this? Thanks a lot!

Your title and question do not match, although they are equivalent. — Andrew Chin, Jan 06 '21 at 04:27
So, instead of solving $x^2 \equiv 28 \pmod 6$, I can solve $x^2 \equiv 4 \pmod 6$. Am I right about this? — , Jan 06 '21 at 04:29
Yes, if $x^2\equiv4$ and $4\equiv28$, then $x^2\equiv28\pmod6$ — J. W. Tanner, Jan 06 '21 at 04:30

score 1 · Accepted Answer · answered Jan 06 '21 at 04:34

1

Since $4\equiv28\pmod6$,

$x^2\equiv4\pmod 6\iff x^2\equiv28\pmod6$.

answered Jan 06 '21 at 04:34

J. W. Tanner

60,406

I got another question is that: Can I solve $x \equiv 4 \pmod 6$ instead of $x \equiv 28 \pmod 6$? Does it applied for non-square number? – Jan 06 '21 at 04:36
Since $4\equiv28\pmod6$, $x\equiv4\pmod6\iff x\equiv28\pmod6$ – J. W. Tanner Jan 06 '21 at 04:37
1

Thank you very much! – Jan 06 '21 at 04:56

score 0 · Answer 2 · 2021-01-07T00:42:06.487

0

We start with $x^2\equiv28\bmod6$. As far as simply checking the two given answers, firstly replace $x$ by $2$ to get the true statement $2^2\equiv28\bmod6$ (since $2^2=4$ and $6|(28-4)=24$).

Similarly, replacing $x$ with $4$ gives us a true statement: $4^2\equiv28\bmod6\iff0\equiv12\bmod6$.

edited Jan 07 '21 at 00:42

answered Jan 06 '21 at 05:15

Understanding the answers of $x^2 \equiv 28 \pmod 6$

2 Answers2