number theory

Find the least positive integer that leaves the remainder 3 when divided by 7, remainder 4 when divided by 9, and remainder 8 when divided by 11

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  1. Hint: Chinese Remainder Theorem

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  2. You can try the CRT (Chinese remainder theorem).

    Master sun proceeded as follows:

    Given
    N≡3 mod 7
    N≡4 mod 9
    N≡8 mod 11
    We need to find the smallest positive N.

    First we find the unit remainders,
    99≡1 mod 7 (99 is product of 9*11_
    154≡1 mod 9 (154 is multiple of 7*11)
    441≡1 mod 11 (441 is multiple of 7*9)

    N mod(7*9*11) is the required number such that
    N=3*99+4*154+8*441 mod 693
    =4441 mod 693
    =283

    Check:
    283≡3 mod 7
    283≡4 mod 9
    283≡8 mod 11 Yeah!

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