# Find the smallest positive integer solution to the following system of congruence: xequiv 17(mod 35) xequiv 8(mod 43)

Find the smallest positive integer solution to the following system of congruence:

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Brighton
Now $x=17+35k$, for some $k\in Z$
Then,

$⇒gcd\left(35,43\right)=1$
Then, there exists x,y \in Z such that
$35x+43y=1$
$\because 35=4×8+3$
$43=35×1+8$
$8=3×2+2$
$3=2×1+1$
$⇒1=3-2×1$
$⇒1=3-\left(8-3×2\right)×1$
$⇒1=3×3-8×1$
$⇒1=3×\left(35-4×8\right)-8×1$
$⇒1=3×35-8×13$
$⇒1=3×35-\left(43-35×1\right)×13$
$⇒1=16×35-43×13$
$⇒1=35\left(16\right)+43\left(-13\right)$
Hence,$x=16$