Fsc Part 1 Mathematics (Complete Solution)

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Q3

Show that the adjointing table is that of multiplication of the elements of the set of residue classes modulo 5.

* |
0 |
1 |
2 |
3 |
4 |

0 |
0 |
0 |
0 |
0 |
0 |

1 |
0 |
1 |
2 |
3 |
4 |

2 |
0 |
2 |
4 |
1 |
3 |

3 |
0 |
3 |
1 |
4 |
2 |

4 |
0 |
4 |
3 |
2 |
1 |

Solution:

The zeros in the second column and the second row are produced by the 0 in the first column and the second row which show that this is a product table, because zero when multiplied any number wil again be zero.Now every element in the table is lesser than 5, so the table is a multiplication table of the elements of the set of residue classes module 5. clearly {0,1,2,3,4} is the set of residues.

thus 3 x 2 = 6 but in place of 6 we insert 1(=6-5).

similarly , in place of 2 x 4 = 8 we insert 3(=8-5) in the table.

* | 0 | 1 | 2 | 3 | 4 |

0 | 0 | 0 | 0 | 0 | 0 |

1 | 0 | 1 | 2 | 3 | 4 |

2 | 0 | 2 | 4 | 6 | 8 |

3 | 0 | 3 | 1 | 4 | 2 |

4 | 0 | 4 | 3 | 2 | 1 |