Worksheet using finiteness

# Worksheet using finiteness - Math 100 Worksheet using...

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Unformatted text preview: Math 100 Worksheet using Finiteness Martin H. Weissman In this worksheet, you will understand and prove the following theorem: Theorem 1. Suppose that p is a prime number. Suppose that a is an integer, and a is not a multiple of p . Then, there exists an integer b , such that ab ≡ 1 (mod p ). 1. Understanding the Theorem Fill in the blanks with integers, in order to make the following sentences true: (1) 3 · ≡ 1 (mod 7). (2) 4 · ≡ 1 (mod 11). (3) 6 · ≡ 1 (mod 31). (4) 20 · ≡ 1 (mod 23). Is it possible to fill in the blanks, in order to make the following sentences true? Why not? (1) 2 · ≡ 1 (mod 4). (2) 7 · ≡ 1 (mod 7). (3) 62 · ≡ 2 (mod 31). 2. Primes as building blocks You can assume the following foundational theorem about prime numbers: Theorem 2. Suppose that p is a prime number. Suppose that a and b are integers. Then, if ab is a multiple of p , then a is a multiple of p or b is a multiple of p ....
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Worksheet using finiteness - Math 100 Worksheet using...

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