## The New Encyclopędia Britannica, Volume 30Encyclopędia Britannica, 1992 - Encyclopedias and dictionaries |

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Page 18

For example , a

satisfies the congruence . This congruence has no

odd for every x and cannot be divisible by 4. But the congruence 2x = 7 ( mod 13

) ...

For example , a

**solution**of the congruence 2x = 7 ( mod 4 ) is any integer x thatsatisfies the congruence . This congruence has no

**solutions**because 2x – 7 isodd for every x and cannot be divisible by 4. But the congruence 2x = 7 ( mod 13

) ...

Page 19

If two or more moduli have a common factor greater than 1 , a

may not exist . An extension of the Chinese remainder theorem states that the

general system will have a

m ; the ...

If two or more moduli have a common factor greater than 1 , a

**solution**may ormay not exist . An extension of the Chinese remainder theorem states that the

general system will have a

**solution**if and only if for every pair of moduli m ; andm ; the ...

Page 23

Therefore , any integer

integer

p the Diophantine equation x2 + y2 = p has integer

mod ...

Therefore , any integer

**solution**of the equation g ( u , v , w ) = 0 leads to aninteger

**solution**of f ( x , y , z ) = 0 . ... According to Euler's theorem , for any primep the Diophantine equation x2 + y2 = p has integer

**solutions**if p = 2 and if p = 1 (mod ...

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### Contents

NUMBER GAMES and Other Mathematical Recreations | 1 |

NUMBER THEORY | 14 |

NUMERICAL ANALYSIS | 38 |

Copyright | |

53 other sections not shown

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