Key to "To determine: The smallest nonnegative integer x that..."

snowlovelydayM

Answered question

2021-05-06

To determine: The smallest nonnegative integer x that satisfies the given system of congruences.

x \equiv 1 \pm o d {4}

x \equiv 8 \pm o d {9}

x \equiv 10 \pm o d {25}

Answer & Explanation

Bentley Leach

Skilled2021-05-08Added 109 answers

$x \equiv 1 \pm o d {4}$
$x \equiv 8 \pm o d {9}$
$x \equiv 10 \pm o d {25}$
First let us solve the first two congruences.
$x \equiv 1 \pm o d {4}$
$x \equiv 8 \pm o d {9}$
We see that the solution x is unique modulo $4.9 = 36$ .
Now, $9 - 4 (2) = 1$ .
Thus,
$x = 1.9 - 8.4 (2)$
$x = 9 - 64$
$x = - 55$
$x = - 19 \pm o d {36}$
$x = 17 \pm o d {36}$
Therefore, $x = 17 \pm o d {36}$ .
Now, let us solve the below equations.
$x \equiv 17 \pm o d {36}$
$x \equiv 10 \pm o d {25}$
We see that the solution x is unique modulo $36.25 = 900$ .
Now, $25 (13) - 36 (9) = 1$ .
Thus,
$x = 17.25 (13) - 10.36 (9)$
$x = 5525 - 3240$
$x = 2285$
$x = 485 \pm o d {900}$
Therefore, $x = 485.$

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