D-Math-W05-Ch5 - SHEN'S CLASS NOTES-191 Chapter Five...

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SHEN’S CLASS NOTES-191 1 Chapter Five Introduction to Number Theory Number Theory is the branch of mathematics concerned with the integers. 5.1 Divisors Definition 5.1.1 Let n and d be two integers, d 0. We say that d divides n if there exists an integer q satisfying n = dq . Moreover, we call q the quotient, and d a divisor (or factor) of n . If d divides n , we write d | n . If d does not divides n , we write d n . Example 5.1.2 3 is a divisor of 21, 3 | 21. 6 is not a divisor of 21, 6 21.
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SHEN’S CLASS NOTES-191 2 Theorem 5.1.3 Let m , n , d be integers. (a) If d | m and d | n , then d | ( m + n ). (b) If d | m and d | n , then d | ( m n ). (c) If d | m , then d | mn . Proof. (a) Because d | m and d | n , we have m = dq 1 , n = dq 2 . Therefore, ( m + n ) = dq 1 + dq 2 = d ( q + q 2 ). Therefore, d | ( m + n ). (b) Similar to (a). ( m n ) = dq 1 dq 2 = d ( q q 2 ). Therefore, d | ( m n ) (c) Because d | m , m = dq . So, mn = dqn = d ( qn ). So, d | mn . Definition 5.1.4 A positive integer p is called prime if p > 1 and its only positive divisors are itself and 1. Otherwise, it is called composite .
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SHEN’S CLASS NOTES-191 3 Example 5.1.5 2, 3, 5, 7, 11, 13, 17, 19, 23 are prime numbers. 34 is a composite number. How to tell an integer n is prime A straightforward way is try integers 2, 3, …, n -1 to see any one of them divides n . If none of them divides n , then it is prime. Example 5.1.6 43 is a prime number because any number in 2, 3, 4, …, 42 can divides 43. Actually, we need not test all these numbers from 2 to n -1. Theorem 5.1.7 A positive integer n > 1 is composite if and only if n has a divisor d such that 2 d n . Proof. We prove the “if” part first. If n has a divisor d such that 2 d n , then n = dq . Because d n < n , d 1 and d n . So, n is composite.
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SHEN’S CLASS NOTES-191 4 Now, we prove the “only if” part. Suppose n is composite. Then n has a divisor c such that 2 c n -1. So, there is an integer q such that n = cq . So, q is a divisor also. Moreover, because 2 c n -1, q 1 and d n . So, 2 q n -1. We claim that either c n or q n . This is because, otherwise, n = cq > n n = n , which is a contradiction. Therefore, n has a divisor d such that 2 d n . Based on Theorem 5.1.7, we have the following algorithm to test if an integer is prime or not. Algorithm 5.1.8 Input: n Output: d Is_prime ( n ) 1 for d = 2 to n 2 if n mod d = 0 3 then return d 4 return 0
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SHEN’S CLASS NOTES-191 5 5 End Example 5.1.9 To check the number 43, we need only to test the following numbers: 2, 3, 4, 5, 6. (6 = 43 ). An interesting fact is that if Algorithm 5.1.8 returns a number d 0, then d must be a prime. Suppose not,
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D-Math-W05-Ch5 - SHEN'S CLASS NOTES-191 Chapter Five...

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