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Question Solve the congruence 2x ≡ 7 (mod 17).

I have tried working out this problem but I am stuck midway. Could someone help me out by showing me or explaining how to proceed further?

Here is the work that I have so far:

Inverse of a modulo m is an integer b for which ab ≡ 1 (mod m), a = 2 m = 17

17 = 8 ⋅ 2 + 1 2 = 2 ⋅ 1 + 0

The greatest common divisor is the last non-zero remainder values, that is, gcd (a, m) = 1.

Expressing the greatest commmon divisor as a multiple of a and m, gcd (a, m) = 1 = 17 - 8 ⋅ 2 = 1 ⋅ 17 - 8 ⋅ 2

The inverse would then be the coefficient of a = 2, which in this case, would be -8. And, since, -8 mod 17 = 9 , 9 is also the inverse of a modulo m.

Solving the congruence 2x ≡ 7 (mod 17) by multiplying each side by the inverse 9, 9⋅2x ≡ 9⋅7 (mod 17) 18x ≡ 63 (mod 17)

And, this is the part where I am stuck. Could anyone help out? Thank you.

Jyrki Lahtonen
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    $18x \equiv x \pmod{17}$, and continue from your last line. – Sarvesh Ravichandran Iyer Oct 19 '20 at 04:59
  • "17 = 8 ⋅ 2 + 1 2 = 2 ⋅ 1 + 0" !!!!PLEASE!!!!! put in line breaks! "seventeen equal eight times two plus twelve equals two times one plus zero"?????? I figure you probably mean $17=8\cdot 2 + 1$ and $2=2\times 1 +0$ but it's really hard to read. – fleablood Oct 19 '20 at 05:31
  • You are done. $18x \equiv 63\pmod {17}$ and $18x \equiv x\pmod {17}$ and so $x \equiv 63\pmod{17}$ is the solution. You are done. If you want to find a number between $0$ and $16$ you know $63 \equiv 12 \pmod {17}$ so $x \equiv 12 \pmod {17}$ is another way of expressing the same solution. ... and to verify $2\times 12\equiv 24 \equiv 7\pmod {17}$ (also $2\times 63\equiv 126 \equiv 7*17 + 7\equiv 7\pmod {17}$. So they are both the same solution. – fleablood Oct 19 '20 at 05:36

3 Answers3

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you are basically done

$18x \equiv 63 \pmod{17}$ and $18x \equiv x \pmod {17}$ and $63\equiv 12 \pmod {17}$

so $x \equiv 12 \pmod{17}$.

fleablood
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$$2x \equiv 7 \pmod {17}$$

$$9\times 2x \equiv 7\times 9 \pmod {17}$$

$$18x \equiv 63 \pmod {17}$$

$$18x \equiv x \equiv 63 \equiv 12 \pmod {17}$$

$63=3\times17+12$

$$x \equiv 12 \pmod {17}$$

Robert Shore
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Lion Heart
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$7+17=24=2 \times 12$, so $x \equiv 12 \pmod{17}$.

Robert Shore
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