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What is the largest four-digit negative integer congruent to \(1 \pmod{23}\)?

 Oct 4, 2020
 #1
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(-x) mod 23 = 1

x =23n + 22, where n =0, 1, 2,3.....etc.

10,000 / 23 =~434

434 - 1 =433

x =23 x 433 + 22 =9,981

(-x) = (-9,981) - the largest negative 4-digit integer.

Because: (-9,981) mod 23 = 1

 Oct 4, 2020
 #2
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Seems the LARGEST negative 4 digit number would be 

-1011              - 1011 mod 23 = 1

Because    -1011 > - 9981       

    in other words  -1011 is larger  than -9981

 Oct 5, 2020

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