1

Solve the equation $285x \equiv 177 \pmod{924}$ using continued fraction

My attempt(using Wikipedia notion):

Continued fraction form for $\frac{924}{285}$ is $[3;4,6,1,9]=[q_1;q_2,q_3,q_4,q_5]$
$\frac{924}{285}=\frac{h_n}{k_n}$

We know that $h_nk_{n-1}-h_{n-1}k_n=(-1)^n \Rightarrow$$924k_{5-1}-285h_{5-1}=(-1)^5$.
Thus, when we find $h_{4}$ we'll get the equation:

$-285h_{4} \equiv (-1)^5 \pmod{924} \Rightarrow$
$ 285h_{4} \equiv (-1)^{4}\pmod{924} \Rightarrow$
$ 285h_{4}(-1)^{4} \equiv 1\pmod{924} \Rightarrow$
$x=h_{4}(-1)^{4}177$ is a solution, because $h_{4}(-1)^{4}$ is $285^{-1}$ modulo 924, thus $285x \equiv 285*285^{-1}b \equiv b \pmod{924}$

Let's find $h_{4}$:

$\frac{h_1}{k_1}=\frac31 \Rightarrow h_1=3$
$\frac{h_2}{k_2}=3+\frac14=\frac{13}4 \Rightarrow h_2=13$
Using $h_n=q_nh_{n-1}+h_{n-2}$:
$h_3=6*13+3=81$
$h_4=1*81+13=94$ Bingo.

Thus $x=94*(-1)^4*175$.

This question is from an exam. The correct answer is $x \equiv 153,461,769 \pmod{924}$

Did I do something wrong? how can $x$ have multiple options(edit: how can I find all the options)?

Any help would be highly appreciated!

miracle173
  • 11,049
Eliran Koren
  • 465
  • 1
    I'd say that this is a strange formulation of the method. The standard formulation is just the extended euclidean algorithm. The long integer divisions and the quotient sequence are of course the same as the one in determining the continued fraction. – Lutz Lehmann Feb 22 '14 at 20:58
  • Except that your quotient sequence is wrong. Should be $[3;4,7,1,2]$, see answer below. – Lutz Lehmann Feb 22 '14 at 21:32
  • There are 3 options, because $\gcd(285,924)=3$, and $3$ divides $177$. – Gerry Myerson Feb 22 '14 at 22:53
  • Oh... I did a stupid mistake. $gcd(285,924)=3 \neq 1$, so I was looking at the wrong equations. I should've done the calculations on the equation $95x=59(mod308)$ – Eliran Koren Feb 23 '14 at 11:07

1 Answers1

2

$285x≡177\pmod{924}$ or $285x+924y=177$ can be solved using the extended euclidean algorithm for the gcd of $285$ and $924$.

The remainder sequence of the euclidean algorithm starts with $r_0=a=924$, $r_1=b=285$ and for the extended variant the Bezout factor sequence for the Bezout identity

$$v_k b≡r_k\;\pmod a\qquad (\text{short version of }u_ka+v_kb=r_k)$$

starts with $v_0=0$, $v_1=1$

Then the algorithm proceeds iterating over $k=1,2,...$

\begin{align} q_k&=r_{k-1}\,\text{div}\,r_k,\\ r_{k+1}&=r_{k-1}-q_kr_k, \\ v_{k+1}&=v_{k+1}-q_kv_k \end{align}

resulting in the table

$$\begin{array}{c|c|c|c|} k& q_k&r_k&v_k\\\hline 0 & & 924 & 0 \\ 1 & 3 & 285 & 1 \\ 2 & 4 & 69 & -3 \\ 3 & 7 & 9 & 13 \\ 4 & 1 & 6 & -94 \\ 5 & 2 & 3 & 107 \\ 6 & & 0 & -308 \\ \end{array}$$

that is, in the end we get the $gcd(285,\,924)=3$ and

$$107\cdot 285 ≡ 3 \pmod{924}.$$

Fortunately, $177=3⋅59$ is divisible by $3$, so $3x≡3⋅59⋅107\pmod{3⋅308}$ has the 3 solutions

\begin{align} x&=59⋅107\mod 308&&=153, \\ x&=153+308&&=461&&\text{ and } \\ x&=461+308&&=769.\end{align}

Lutz Lehmann
  • 126,666