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# Number theory...

0
5
3
+38

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.

Aug 6, 2024

#1
+1323
-1

Using modular arithmetic, the answer is 3.

Aug 6, 2024
#2
+1790
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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! :)

Aug 9, 2024
edited by NotThatSmart  Aug 9, 2024