+50 Let x be an integer such that \(x \equiv 5 \pmod{7}\) . Find the value of .\(2a + 12 \pmod{7}\) Express your answer as a residue between 0 and the modulus.
+1908 Hi UnVerifiedTaxPayer!
ummm..just a note, but you used different variables in the problem, \(a\) and \(x\)
I'm assuming this is a typo, but if it isn't correct me. Let's just say we are trying to find the value of \(2x + 12 \pmod{7}\)
Now, since we already have the first equation in handy, we can easily do a simple process.
Let's try to turn the left side of \(x \equiv 5 \pmod{7}\) into 2x+12.
First, let's multiply both sides of the equation by 2. This gets us
\(2x \equiv 10 \pmod7\)
Since 10 mod 7 is the same as 3 mod 7 (they both leave a remainder of 3), we have the equation \(2x \equiv 3 \pmod 7\)
Now, let's add 12 to both sides of the equation. We then get \(2x+12 \equiv 15 \pmod7\)
Since 15 mod 7 is the same as 1 mod 7, we are left with
\(2x+12 \equiv 1 \pmod7\)
Now, we can check our answer.
Take 5 as an example. We have that
\(5 \equiv 5 \pmod7\)
We also have that \(2(5) + 12 = 22 \equiv 1 \pmod7\)
This also works for 12, 19, and 26, so our answer should be correct.
So our answer is 1. Again, correct me if I misunderstood the typo in the problem!
Thanks! :)
