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Multiplication Modulo p

The multiplication modulo

where

For example,

Surely, the set

Hence

**p**,**p**being a positive integer, of any two integers**a**and**b**is denoted by**a ×**and is defined as_{p}b**a ×**,_{p}b = r, 0 ≤ r pwhere

**r**is a least non-negative remainder when**ab**(i.e., the product of**a**and**b**) is divided by**p**. In other words, we find an ordinary product of two integers**a**and**b**, viz.**ab**and from this product we remove the multiples of**p**in such a way that the remainder**r**left out is such that either**r = 0**or**r < p**and**r**is a positive integer.For example,

**8 × 5 7 = 1 as 8 × 7 = 56 = 5.11 + 1**

9 × 4 7 = 3 as 9 × 7 = 63 = 15.4 + 3.9 × 4 7 = 3 as 9 × 7 = 63 = 15.4 + 3.

**Example:**Prove that the set**{1, 2, 3, 4, 5, 6}**is a finite Abelian group of order**6**under multiplication modulo**7**.**Solution:**Let**G = {1, 2, 3, 4, 5, 6}**and let us form the composition table**X**_{7},X_{7} |
1 | 2 | 3 | 4 | 5 | 6 |

1 | 1 | 2 | 3 | 4 | 5 | 6 |

2 | 2 | 4 | 6 | 1 | 3 | 5 |

3 | 3 | 6 | 2 | 5 | 1 | 4 |

4 | 4 | 1 | 5 | 2 | 6 | 3 |

5 | 5 | 3 | 1 | 6 | 4 | 2 |

6 | 6 | 5 | 4 | 3 | 2 | 1 |

**G**From the composition table, it is clear that all the entries are the element of_{1}: Closure property:**G**and, therefore, closure property holds good.**G**For all_{2}: Associative law:**a, b, c G**,**aX**_{7}(bX_{7}c) = a ×_{7}(b.c) as b ×_{7}c≡bc (mod**7)****=**least non-negative remainder obtained on dividing**a(bc)**by**7**.**=**least non-negative remainder obtained on dividing**(ab)c**by**7**.**[∵ (ab)c = a. (bc) ∀ a, b c I]****= (cb) × 7 c****≡ (a × 7b) X**_{7}c [∵ aX_{7}b≡ab mod 7]**G**is the identity element as_{3}: Identity element: 1 G**1 × 7 a = a = aX**_{7}1**G**The inverses of_{4}: Existence of inverse:**1, 2, 3, 4, 5, 6**are**1, 4, 5, 2, 3, 6**respectively.**G**The corresponding rows and columns in the composition table are identical, and as such the commutative law holds good._{5}: Commutative law:Surely, the set

**G**has**6**elements.Hence

**<G, × 7>**is a finite albelian group of order**6**.**Services: -**Multiplication Modulo p Homework | Multiplication Modulo p Homework Help | Multiplication Modulo p Homework Help Services | Live Multiplication Modulo p Homework Help | Multiplication Modulo p Homework Tutors | Online Multiplication Modulo p Homework Help | Multiplication Modulo p Tutors | Online Multiplication Modulo p Tutors | Multiplication Modulo p Homework Services | Multiplication Modulo pSubmit Your Query ???

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