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.

AUnVerifiedTaxPayer Aug 6, 2024

#2**+1 **

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! :)

NotThatSmart Aug 9, 2024