To determine: The smallest nonnegative integer x that satisfies the given system of congruences.x\equiv 3\pmod 5x\equiv 3\pmod 7

To determine: The smallest nonnegative integer x that satisfies the given system of congruences.
$x\equiv 3±5$
$x\equiv 3±7$

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Derrick

$x\equiv 3±5$
$x\equiv 3±7$
We see that the solution x is unique modulo 7.5=35
Now, 5(10)-7(7)=1.
Thus,
x=4.5(10)-3.7(7)
x=200-147
x=53
$x=18±35$
Therefore, x = 18.